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I'm working on Isaacs' Character Theory For Finite Groups, 11.17. It asks you to show that $|M(G)|=2$, where $G=A_5$ and $M(G)$ is the Schur multiplier of $G$.

Here're some of my thoughts:

By Corollary 11.21, $M(G)$ is a 2-group. Let $(\Gamma,\pi)$ be a Schur representation group for $G$ such that $\ker(\pi)=M(G).$ Let $P\in\mathrm{Syl}_2(G)$. Let $H$ be the inverse image of $P$ under $\pi$. Then $H/M(G)=P\cong\mathbb{Z}_2\times\mathbb{Z}_2$.

I want to show that $M(G)\leq H'\cap Z(H).$

If this were shown, then $M(G)$ is isomorphic to a subgroup of $M(\mathbb{Z}_2\times\mathbb{Z}_2)$ by Corollary 11.20. By Problem 11.16, $|M(\mathbb{Z}_2\times\mathbb{Z}_2)|=2$. Thus $|M(G)|\leq 2.$

To show $M(G)\neq 1$, I want to find a nontrivial extension of $\mathbb{Z}_2$ by $A_5$, but I have no idea.

abvdd
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    Related question https://math.stackexchange.com/questions/3536218/prove-that-g-sl2-mathbbf-5-is-an-extension-of-mathbbz-2-by-a-5?rq=1 will answer your second question. – ancient mathematician Jun 29 '21 at 08:44

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For the first question note that $\Gamma/(H' \cap Z(\Gamma))$ has abelian Sylow $2$-subgroups, so you can use Thm 5.6 to deduce that its centre is trivial, and hence $Z(\Gamma) < H'$.

For the second question, the group ${\rm SL}(2,5)$ is the required nontrivial extension.

Derek Holt
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