Decomposition of axial vector and vector representions of C4v group

Let $R$ be the orthogonal matrix corresponding to an operation in $O(3)$. If
R is a proper rotation, then both vectors $\vec{V}$ and axial vectors $\vec{A}$ are transformed in the same way


$$ \vec{V} \rightarrow \vec{V}' = R\vec{V}$$
$$ \vec{A} \rightarrow \vec{A}' = R\vec{A}$$


if $R$ is an improper rotations, then
$$ \vec{V} \rightarrow \vec{V}' = R\vec{V}$$
$$ \vec{A} \rightarrow \vec{A}' = -R\vec{A}$$


And let $\mathcal{V}$ be the vector representation of $O(3)$ and $\mathcal{P}$ be the axial-vector representation of $O(3)$. Thus we can write,
$$\vec{V} \rightarrow \vec{V}' = \mathcal{V}(R)\vec{V}$$
$$\vec{A} \rightarrow \vec{A}' = \mathcal{P}(R)\vec{A}$$


My question is, considering $\mathcal{V}$ and $\mathcal{P}$ as representations of $C\_\mathrm{4v}$ group, how can I find their composition in terms of irreps ?


$$\begin{array}{c|ccccc|cc} \hline
C\_\mathrm{4v} & E & 2C\_4 & C\_2 & 2\sigma\_\mathrm{v} & 2\sigma\_\mathrm{d} & & \\ \hline
\mathrm{A\_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\
\mathrm{A\_2} & 1 & 1 & 1 & -1 & -1 & R\_z & \\
\mathrm{B\_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\
\mathrm{B\_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ 
\mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R\_x,R\_y) & (xz,yz) \\ \hline
\end{array}$$


