= Faithful four-point action of GL2 over F2
{title2=$GL_2(\mathbb F_2)\cong S_3$}
The <general linear group over a finite field> $GL_2(\mathbb F_2)$ acts faithfully on the four <vectors> of $\mathbb F_2^2$, since a matrix fixing all vectors fixes the standard basis. The zero vector is fixed by every element; the induced action on the three nonzero vectors identifies this six-element group with $S_3$. Swapping the two coordinates fixes $(0,0)$ and $(1,1)$ and exchanges the other two vectors, so its four-point <permutation> is odd. This action therefore embeds the group into $S_4$ with nontrivial composite <sign homomorphism>.
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