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Given an abelian group $M$. First I'd like to know if $\text{End}(M)$ is generated by $\text{Aut}(M)$ (as ring, or equivalently, as additive group). Second I'd like to know if it doesn't hold generally, then what condition will guarantee it.

I guess it holds at least when $M/\text{Tor}(M)$ is a free abelian group. And under this constraint, it suffices to consider abelian $p$-groups and free groups. I succeeded to prove that it always holds for free abelian groups of finite rank. But I cannot prove the case of abelian $p$-groups and general free abelian groups.

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It's not true for $M=C_2\times C_4$.

Let $x$ and $y$ be generators of the two factors. Then every endomorphism $\varphi$ is of the form $$\begin{align} \varphi(x)&=ax+2by\\ \varphi(y)&=cx+dy \end{align}$$ for integers $a,b,c$ determined$\pmod{2}$ and $d$ determined$\pmod{4}$. But if $\varphi$ is an automorphism then $\varphi(y)$ has order $4$, and so $d$ is odd. But then $a$ is also odd, or else $\varphi(x)=2by$ is a multiple of $\varphi(y)$.

So for all automorphisms, and hence all endomorphisms that are sums of automorphisms, $a+d$ must be odd. So in particular the projection onto either factor is not in the subgroup of $\operatorname{End}(M)$ generated by $\operatorname{Aut}(M)$.

But this example seems very special to $p=2$

EDIT: There seems to be quite a lot of literature on this problem going back to the 1950s. See for example this 2006 paper by C. Meehan, especially the introduction and references. Some things that seems to be known: free abelian groups of arbitrary rank have this property (in fact, every endomorphism is the sum of at most two automorphisms), and finite abelian $p$-groups for $p>2$ have the property. It doesn't seem to be true for abelian $p$-groups in general, even when $p>2$, but there is a property ("total projectivity") that implies it for $p>2$.

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    $\begingroup$ An unexpected example... Furthermore I found that the same argument applies to all the group of form $C_{2^a}\times C_{2^b}\times N$, where $a\neq b\in\mathbb N^+$ and $N$ a finite $2$-group such that no $a$ and $b$ doesn't occur in its set of elementary factors... $\endgroup$ – Censi LI Mar 18 '15 at 18:05

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