I am in search of some examples of finite groups $G$ and modules over the ring $\mathbb{Z}[G]$ with following conditions (and examples taken independently). I do not have idea whether such examples are possible, and the question can be very trivial; but I didn't immediately get example(s) to justify some statements in an introduction to integral representations in Curtis-Reiner's representation theory book.
1) Finite group $G$ and a $\mathbb{Z}[G]$-module $M$ such that $M$ has no composition series. [This implies that Jordan-Holder theorem do not hold for $M$; am I right?]
2) Finite group $G$ and a $\mathbb{Z}[G]$-module $M$ such that Krull-Schmidt theorem fails for $M$.
3) Finite group $G$ and a $\mathbb{Z}[G]$-module $M$ such that $M$ contains a proper non-zero $\mathbb{Z}[G]$-submodule but has no complement (i.e. that submodule is not a direct summand).
If $\mathbb{Z}$ is replaced by a field $K$ whose characteristic does not divide $|G|$ then above examples are not possible- standard beginning of representation theory of groups. But Curtis-Reiner says that such results (Jordan-Holder/ Krull-Schmidt/ Maschke's theorem) are no longer true if the field $K$ is replaced by a ring $R$. In this regard, I was looking for simple examples for failure.