This is question related to following link:
Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$
$M$ and $N$ as in the linked post, that is, $M=\mathbb{Z}$ an $N=\bigoplus_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$.
So you constructed an ses $S_1: 0\longrightarrow M\stackrel{\alpha}\longrightarrow M\oplus N\stackrel{\beta}\longrightarrow M\longrightarrow 0$ that is not split. Thus $S_1$ cannot be equivalent with $S_0: 0\longrightarrow M\stackrel{i}\longrightarrow M\oplus N\stackrel{\pi}\longrightarrow M\longrightarrow 0$ in which $i$ and $\pi$ are the natural injection and natural projection, respectively. Therefore, there cannot be an isomorphism $\phi : M\oplus N \longrightarrow M\oplus N$ such that $\phi \circ \alpha = i$.
Is it possible to prove that the identity is the only isomorphism $M\oplus N \longrightarrow M\oplus N$ ? I will be grateful if you / someone can prove that for me.
Your question is whether the $\mathbb{Z}$-module $\mathbb{Z}\oplus \bigoplus_{\mathbb{N}} \mathbb{Z}/2\mathbb{Z}$ admits automorphisms other that the identity.
As Hagen von Eitzen says in a comment, any non-trivial automorphism of $\mathbb{Z}$ or of $\bigoplus_{\mathbb{N}} \mathbb{Z}/2\mathbb{Z}$ will give rise to such an automorphism. Since $\mathbb{Z}$ has a non-trivial automorphism (given by multiplication by $-1$), this answers your question.
Note that $\bigoplus_{\mathbb{N}} \mathbb{Z}/2\mathbb{Z}$ has even more non-trivial automorphisms.