Extension of a p-group abelian by simple group

Question

Let $N$ be finite abelian $p$-group (is not cyclic). Is there any extension (not central extension) of $N$ by $A_{5}$?

Answer 1

Here's a big database of them.

Answer 2

The easiest way to construct such examples is to take a nonzero irreducible module $M$ for $A_5$ over the field of order $p$, and let $G$ be the semidirect product of $M$ by $A_5$.