Let $A$ be the set of these matrices and their permutations: $$ \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{bmatrix}, \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0& 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix}, \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$ So each has three permutations $P_1, P_{123}, P_{132}$ so in total there are 12 elements, so the order of $A$ is 12.
$A_4$ is the subgroup of even permutations in $S_4$. $A_4$ is also of order 12.
I get the intuition of how they are isomorphic, since $P_1, P_{123}, P_{132}$ are all even permutations, and both $A$ and $A_4$ are of order 12. So it would clearly be one to one and onto. I'm having trouble formally expressing how this represents an isomorphism.