Two years ago, Justice Anthony M. Kennedy played down the impact of the decision he had just joined in Hobby Lobby v. Burwell that relieved religiously objecting owners of certain businesses from providing contraceptive coverage to their employees.

It was the Supreme Court’s third consideration of the mandates placed on employers under the 2010 Affordable Care Act, and three justices had joined Justice Ruth Bader Ginsburg’s scorching dissent.

The court was in agreement that the government’s requirement “furthers a legitimate and compelling interest in the health of female employees,” Kennedy said. The question was whether it could be carried out without infringing on religious freedoms of employers who must provide the coverage.

The solution Kennedy suggested — an accommodation that would insulate employers from providing the contraceptive coverage but still ensure that their employees receive it — will be at the heart of the discussion Wednesday when the Supreme Court undertakes its fourth consideration of what is popularly called Obamacare.

“As usual, all eyes will be on Justice Kennedy,” Elizabeth B. Wydra, president of the left-leaning Constitutional Accountability Center, said during a discussion of the case last week at the Cato Institute.

This case involves not private employers but religiously affiliated organizations such as universities, hospitals and charities.

The Obama administration says it has provided the organizations with an easy way out, just as Kennedy suggested. Employers who object must make their religious objections clear by signing a form or sending a letter and let insurance companies and the government take over from there.

But the groups say that even that step would implicate them in sin and that they face ruinous fines if they refuse to comply. They want to be included under the same blanket exemption from providing the coverage that the government has extended to churches and other purely religious groups.

The court accepted seven cases from throughout the country, including one challenge involving the Roman Catholic Archdiocese of Washington and another from an order called the Little Sisters of the Poor, which runs homes for the elderly.

Conflicting lower-court decisions resulted in supporters and opponents of the law calling for the Supreme Court to act. All but one of the nation’s regional courts of appeal have ruled in favor of the government.

With Justice Antonin Scalia’s death, eight justices will hear the case. It seems unlikely that the court’s liberal objectors in the Hobby Lobby case — Ginsburg and Justices Stephen G. Breyer, Sonia Sotomayor and Elena Kagan — will not find the accommodation sufficient.

If they win Kennedy’s support, the law’s requirements will be in place nationally. But a 4-to-4 tie would mean the mandate would be carried out only in those regions of the country where courts have ruled for the government, and not in the other.

A deadlocked court could also schedule a rehearing when the court has a ninth member, but no one knows when that might be.

The case pits questions of religious liberty against a woman’s right to equal health-­care access.

U.S. Solicitor General Donald B. Verrilli Jr. told the court in the government’s brief that the organizations want more than to be left alone.

“They assert a right not only to be relieved of the obligation to provide contraceptive coverage themselves, but also to prevent the government from arranging for third parties to fill the resulting gap,” Verrilli wrote. “If accepted, that claim would deny tens of thousands of women the health coverage to which they are entitled under federal law, and subject them to the harms the law is designed to eliminate.”

Catholic organizations that brought the lawsuits say it is incorrect for the government to say the accommodation allows them to opt out of providing the coverage.

“Quite the opposite: the government is forcing petitioners to take actions that cause the objectionable coverage to be delivered to petitioners’ own employees and students by petitioners’ own insurance companies in connection with petitioners’ own health plans,” said the brief filed by Washington lawyer Noel J. Francisco.

Added Lori Windham, senior counsel at the Becket Fund for Religious Liberty, which has represented many of the challengers: “This is a question of theology and morality.”

As in Hobby Lobby, the complaint is that the contraceptive mandate promulgated by the Department of Health and Human Services violates the Religious Freedom Restoration Act.

The RFRA says the government must have a compelling reason for laws and programs that substantially burden religious beliefs, and even then government must prove that the law is the least burdensome way of achieving its goal.

In ruling for Hobby Lobby, the court’s conservatives suggested that one reason the business owners in that case had a valid complaint was that the government had made special arrangements for churches and religious nonprofits but not for them.

In the current litigation, most appeals courts have ruled that the government work-around suffices.

“All plaintiffs must do to opt out is express what they believe and seek what they want via a letter or two­-page form,” Judge Cornelia T.L. Pillard said when the case involving the Washington archdiocese came before the U.S. Court of Appeals for the District of Columbia Circuit. “Religious nonprofits that opt out are excused from playing any role in the provision of contraception services, and they remain free to condemn contraception in the clearest terms.”

But several prominent conservative judges have protested the rulings, and in September, the U.S. Court of Appeals for the 8th Circuit, in St. Louis, became the first to rule against the government.

In a case involving a college and a religious charitable organization, Judge Roger L. Wollman wrote for a unanimous appellate panel that the issue is whether the groups “have a sincere religious belief that their participation in the accommodation process makes them morally and spiritually complicit in providing abortifacient coverage. Their affirmative answer to that question is not for us to dispute.”

The cases accepted are Zubik v. Burwell, Priests for Life v. Department of HHS, Roman Catholic Archbishop v. Burwell, East Texas Baptist University v. Burwell, Little Sisters of the Poor v. Burwell, Southern Nazarene University v. Burwell, and Geneva College v. Burwell.