Father put husbands name(son) in the deed when he invested $$ in the home. When they die he inheriates thehome. Father since died, step mom lives there and now has moved in with her daughter. Can she leave the home abanddant or rent it out without his permission.

What r deed owners rights 2 home with step mom on there till she passes. She has moved out, can she leave the home abandant or rent it out?

You have to write your question more clearly if you want us to understand you. "when they die he inheriates thehome." We don't even know who THEY DIE refers to, who is they? " father since died, step mom lives there and now has moved in with her daughter" did he will the home to her, was there a will?

father put husbands name(son) in the deed when he invested $$ in the home. when they die he inheriates thehome. father since died, step mom lives there and now has moved in with her daughter. can she leave the home abanddant or rent it out without his permission.

The late father conveyed the property to his son? If that is what your saying, the remainder of your sentence, "when they die he
Inheriates [should be spelled "inherits"] the home", doesn't make any sense.

So perhaps you mean that

the father and his wife conveyed the property to the son reserving a joint life estate to the father and father's wife (son's step-mom)? or
father and his wife own the home in common; father conveys his (father's) interest to the son; father dies, leaving step-mom with a one half undivided interest in the home?

If so, the step-mom would continue to have a life estate (or a 1/2 undivided interest). Yes, she can abandon it, or (in the case of a life estate) rent it out without the son's permission. If it's the half interest scenario, he can sue her for partition.

If it's something else, please explain.

I'm guessing that the son on the deed (poster's husband) wants to know what his rights are if his step mother moves out. He can sue for partition as AKlawyer says, as I understand it. That means he has to go to court to get the right to put it up for sale.