$A_5$ as a unique homomorphic image Is there an infinite group whose only (nontrivial) finite homomorphic image is $A_5$ (the alternating group of degree $5$) ? **Edit** : I am interested in a group that is not the direct product of $A_5$ and an infinite group without finite quotients.

I think the following will work. Let $M$ be an irreducible module for $A_5$ over ${\mathbb Q}$. For example, we could take $M = {\mathbb Q}^4$ to be the 4-dimensional deleted permutation module. Now let $G = M \rtimes A_5$ with the module action of $A_5$ on $M$.