In a first version of this answer  I claimed incorrectly (see Francesco's answer), that there are no examples inside smooth simply connected threefolds. In fact what I had in mind is the well known fact that  an Enriques surface cannot be a hyperplane section of a smooth  threefold.
I leave here the closing remark, because it is perhaps useful. 

Concerning the example of the Fermat quartic, notice that an involution of $\mathbb P^3$ has fixed points on any surface $S\subset \mathbb P^3$ so you cannot get a free involution this way. Blowing up does not improve things, because you replace a fixed point wih a curve all made of fixed points.