authenticate members of listserve

I would like to control access to a wordpress site. I want to let members of a specific google group and a specific yahoo group into the site, but no one else. I am making this wordpress site for the members of two listserves that include some privacy fanatics.

I do not want to password protect the pages, since a number of the members of these pages would find that too daunting.

Each of these listserves has a moderator who keeps the subscriptions up to date. The groups are for parents of specific schools, and each year there is significant turnover. I do not want to duplicate the subscription maintenance that is already being done.

To put this into other words, here is what I want to happen:
– User Joe is a member of his school’s listserve. He happens to be logged into whatever account he used to join the listserve. He goes to the link I have sent to his email via his school’s listserve, arrives safe and sound at my wordpress site and navigates to his heart’s content.
– User Mary is, like Joe, a member of her school’s listserve, but unlike Joe, she doesn’t happen to be logged in. When she arrives at my url, she is instructed to log in to the email account she used to create her listserve subscription. She gets delivered happily to my site.
– Darth is logged into his email account of choice, but is not a member of either school’s listserve. He somehow obtains the link to my wordpress site, but when he gets there, he gets a message saying he is not authorized to view the page. He is told what hoops to go through to get access.

I wonder if OpenID would be helpful for this, and if so, where to go for guidance with that.

Note that Google Drive and Googlesites already have this way of authenticating built in, at least for members of a google group. I would like to make an attractive site, i.e. a wordpress site. I think googlesites sites look tacky.

I hope someone can help. Thank you!