WATCH: Liz Cheney Reiterates Opposition to Marriage Equality

Liz Cheney, in a television interview Sunday, reiterated her opposition to marriage equality, in light of allegations that she actually supports it -- and she later got some online criticism from her lesbian sister.

On the Fox News Sunday program, host Chris Wallace asked Cheney, who is seeking the Republican U.S. Senate nomination in Wyoming, if their was some conflict in stands she had taken regarding the issue. She has opposed a federal constitutional amendment to ban same-sex marriage and supported the State Department's offering of benefits to same-sex partners of employees, but she has also said she endorses "the traditional definition of marriage."

"I stand by both of those positions," Cheney, daughter of former vice president Dick Cheney, told Wallace. "I don't believe we ought to discriminate against people because of their sexual orientation. If people are in a same-sex relationship and they want their partner to be able to have health benefits or be designated as a beneficiary in your life insurance, there's no reason we shouldn't do that. I also don't support amending the Constitution on this issue. I do believe it's an issue that's got to be left up to states. I do believe in the traditional definition of marriage."

Cheney also said she won't change her position just because her sister, Mary Cheney, is married to a woman. I love Mary very much. I love her family very much," Liz Cheney said. "This is just an issue on which we disagree."

Mary Cheney responded by writing on her Facebook page, "Liz -- this isn't just an issue on which we disagree -- you're just wrong -- and on the wrong side of history." She has objected to her sister's previous comments on the issue as well.

Mary's wife, Heather Poe, also responded on Facebook, saying, "Liz has been a guest in our home, has spent time and shared holidays with our children, and when Mary and I got married in 2012 -- she didn't hesitate to tell us how happy she was for us. To have her now say she doesn't support our right to marry is offensive to say the least. I can't help but wonder how Liz would feel if as she moved from state to state, she discovered that her family was protected in one but not the other. I always thought freedom meant freedom for EVERYONE."

Liz Cheney is running against incumbent senator Mike Enzi in the Republican primary. A recent poll commissioned by a conservative super PAC supporting Enzi indicates she has little chance of unseating him; the poll showed him leading Cheney 69 percent to 17 percent among likely primary voters, according to Politico. Enzi's supporters have criticized Cheney as insufficiently conservative on social issues, including marriage equality, and various political observers have questioned if she should even be running in Wyoming, the state her father represented in Congress some years ago, as she has long been a resident of Virginia.

Watch video of Liz Cheney on Fox News Sunday below, courtesy of The Raw Story.