share or shares

One of the most popular social networking sites is Facebook. It lets people update their personal profiles and shares the updates with otherpeople who they have declared to be their friends.
I think the subject of the verb “share” is “people”. Therefore, it should be “share” not “shares”. Is it correct?
In the sentence the verb “share” has a single form. It make me think that the subject of the verb is “it” or “Facebook” but I have a feel that this is incorrect.

It’s Facebook that shares the profiles. The subject is “it”.

Thank you!
I thought that but it seems to me that only people can share their profiles and Facebook, by itself, can’t share the profiles.

Facebook can share the profile in your absence. In fact, it can share that information after you’ve died and been buried.