How to construct an injective homomorphism from $GL(2,\mathbb{Z}_2)$ into $S_4$?

Question

I am struggling to devise an injective homomorphism from $GL(2, \mathbb{Z}_2)$ to $S_4$, in particular construction which actually allows me to verify whether its a homomorphism or not.

Eg, we could map as follows. Map a matrix $M$ with coefficients in $\mathbb{Z}_2$ to the cycle $(M_{11}, M_{12}, M_{21}, M_{22})$.

It's very hard, however, to verify that this is a homomorphism. It's also not injective. I've also considered double-row notation, but I don't think that this works either.

Answer 1

$GL(2,\mathbb Z_2)$ acts on $\mathbb Z_2 \times \mathbb Z_2$ by matrix vector multiplication. This action is faithful, hence induces an injective homomorphism $GL(2,\mathbb Z_2) \to S(\mathbb Z_2 \times \mathbb Z_2) \cong S_4$.

We say an action of $G$ on $X$ is faithful, if $g.x = x$ for all $x \in X$ only happens for $g =1 \in G$.

I.e. any matrix $A \in GL(2,\mathbb Z_2)$ permutes all vectors in $\mathbb Z_2 \times \mathbb Z_2$. As there are 4 vectors and no matrix but the identity leaves all vectors unchanged, this gives an explicit injective homomorphism to the symmetric group on 4 elements.

Answer 2

Since $GL(2,\Bbb Z_2)$ is isomorphic to $S_3$, simply extend the isomorphism there to an injective homomorphism from $GL(2,\Bbb Z_2)$ to $S_4$, which can be done as there exists an $H\le S_4$ such that $S_3\cong H$.