Stronger version of Mackey's decomposition theorem I would like to know if the following result is correct:

Let $H$ and $G'$ be subgroups of a finite group $G$, $W$ be a $\Bbb C$-representation of $H$, and $N$ be a normal subgroup of $G'$.
  Then
  $\renewcommand{\Res}{\mathrm{Res}}
\renewcommand{\Ind}{\mathrm{Ind}}$
  $$
[ \Res_{G'}( \Ind_H^G(W)) ]^{ N }
\;\;=\;
\bigoplus_{x \in G' \;\setminus G \;/ H}  
\left(       \mathrm{Ind}_{G'  \;\cap\; xHx^{-1}}^{G'} \;\;(xW)     \right)^{N}
$$

where $(\rho_x,xW)$ is the representation of $xHx^{-1}$ given by
$\rho_x(y)=\rho(x^{-1}yx)$
and
for a given representation $(\rho',V')$ of $G'$,
$$(V')^N := \{v \in V'   \;\mid\;   \rho'(n)v=v,    \;\forall n \in N\}.$$

Remark:
Since $N$ is normal in $G'$, the representation $$\widetilde{\rho'} : g' \mapsto \rho'(g')\Big\vert_{(V')^N}$$
is well-defined, i.e. $(V')^N$ is a $\Bbb C[G']$-module (or: $(V')^N$ invariant under $G'$).
My thoughts:
I think that it should follow directly from Mackey’s Restriction Formula (e.g. theorem 12.4 here) and from the fact that
$$(V_1 \oplus V_2)^N = V_1^N \oplus V_2^N$$
where $V_j$ are $\Bbb C[G']$-modules.
However, I'm not quite sure because I've seen in some reference that
$$
[ \Res_{G'}( \Ind_H^G(W)) ]^{ N }
\;\;=\;
\bigoplus_{x \in G' \;\setminus G \;/ H}  
\left(       \mathrm{Ind}_{G'  \;\cap\; xHx^{-1}}^{G'} \;\;(xW)     \right)^{N  \;\cap\;  xHx^{-1}}
$$
They have $(\cdots)^{N  \;\cap\;  xHx^{-1}}$ at the end, instead of just $(\cdots)^N$ as I think it should be.

After some thinking, I'm sure that the first version is correct but the second is not. We must indeed take the subspaces invariant under $N$ both sides.
