Is there a non-split central extension of an infinite cyclic group by a finite simple non-abelian group?
I have tried, very naïvely, I know, to just take a presentation of $A_5$, say $\langle x,y|x^2=y^3=(xy)^5=1 \rangle$, and then change it this way $\langle x,y,z|x^2=y^3=(xy)^5=z \rangle$. As you probably would see at first glance, this way one ends up with a non-split central extension of a finite(!) group by $A_5$.
So, is there a way of getting such extension? Is there a way of modeling presentations in a wiser manner to get what we want?
Let $E$ be a group with infinite cyclic normal subgroup $N$ and $G/N$ a finite nonabelian simple group (or any finite perfect group). Since $|E:Z(E)|$ is finite, a theorem of Schur says that the commutator subgroup $[E,E]$ is finite, and hence $[E,E] \cap N = 1$.
But $E/N$ perfect implies $E = [E,E]N$, so $E = [E,E] \times N$ is a split extension, and the answer to your question is no.