= Solution
The classes of $A_4$ are the identity, the three double transpositions, and two classes of four three-cycles. Its quotient by $V_4$ is cyclic of order three, giving <characters> $\mathbf1,\chi,\chi^2$, with $\chi$ taking values $1,1,\omega,\omega^2$ on these classes. The restriction of $V$ has <character> $(3,-1,0,0)$ and <norm> $(9+3)/12=1$, so it is the three-dimensional <irreducible representation> $T$. These four <representations> are complete since $1+1+1+9=12$.
The sign is trivial on $A_4$, so
$$
\boxed{\operatorname{Res}\mathbf1=\operatorname{Res}\mathrm{sgn}=\mathbf1,\quad \operatorname{Res}V=\operatorname{Res}(V\otimes\mathrm{sgn})=T,\quad \operatorname{Res}W=\chi\oplus\chi^2.}
$$
For the last equality the restricted <character> is $(2,2,-1,-1)$, the sum of the two indicated <one-dimensional characters>. Equality of <characters> proves the decomposition.
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