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WASHINGTON (CNN) -- Four of the 17 Republican presidential candidates have pledged to support a federal constitutional amendment defining marriage as a union between one man and one woman, an anti-same-sex marriage group said Tuesday.

The National Organization for Marriage said Texas Sen. Ted Cruz, former Pennsylvania Sen. Rick Santorum, Louisiana Gov. Bobby Jindal and retired neurosurgeon Ben Carson signed the group's pledge "to take several specific actions as president to restore marriage to the law and protect people of faith from discrimination because of their support for traditional marriage."

The influence of the National Organization for Marriage -- once a powerful force in funding pushes for same-sex marriage bans on the state level -- has been in decline following a large swing in public support in favor of same-sex nuptials. According to Gallup, in 1996 nearly seven in 10 Americans opposed same-sex marriage, but in May 2015, 60% of Americans said they supported such unions.

And in June, the Supreme Court ruled 5-4 that states must recognize same-sex marriage.

"Electing a president in 2016 who is a true champion on marriage is a critical priority, something that is essential if we are to overturn the outrageous, illegitimate decision of the U.S. Supreme Court imposing same-sex 'marriage' on every state in the nation," Brian Brown, president of the National Organization for Marriage, said in a statement.

In addition to support for the federal constitutional amendment, the group's pledge includes working to overturn the Supreme Court decision, nominating judges and appointing attorney generals who will apply "the original meaning of the Constitution," and reviewing pro-gay rights policies enacted by the Obama administration.

The pledge also asks potential presidents to direct the Justice Department to investigate and publicize cases of Americans who have been harassed and threatened due to their belief that marriage is between one man and one woman.

NOM invited every candidate to sign, and Brown said the organization will not support a candidate who does not sign the pledge.

Wisconsin Gov. Scott Walker and former Arkansas Gov. Mike Huckabee -- both courting evangelical voters -- have spoken frequently about their support for traditional marriage, but declined to sign the pledge, Brown said in the release.

Both men told the group they have a blanket policy against signing pledges, Brown added.

Several GOP candidates, including former Florida Gov. Jeb Bush, South Carolina Sen. Lindsey Graham and Ohio Gov. John Kasich, have stated their opposition to same-sex marriage but do not support amending the Constitution to make marriage between one man and one woman, they have said.

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