Finite $\Delta$-module of $p$-power order

Finite $\Delta$-module of $p$-power order

I have a question concerning lemma, that I want to prove:
Let $p$ be a prime and $\Delta$ be a finite group of order prime to $p$.
Let $M$ be a finite $\Delta$-module of order a power of $p$. Then there is
an isomorphism of $\mathbb{F}_p[\Delta]$-modules $$_p M \cong M/pM$$ where
$_p M$ denotes the kernel of the multiplication by $p$.
My problem now is that I'm able to prove the above theorem in the case of
$\mathbb{Z}$-modules. But in our case - that of
$\mathbb{Z}[\Delta]$-modules - I'm totally lost. In the case of
$\mathbb{Z}$-module I would have just used the decomposition theorem for
abelian groups. Can you help me with the case of group rings??
For example the module $\mathbb{Z}/p^2 \mathbb{Z}$ is of order $p^2$ and
let the action of $\Delta$ be trivial. How do I get the above isomorphism?
Thank you for your help, Tom