Let $ \Gamma $ be a group which fits into a SES
$$
1 \to \mathbb{Z}^n \to \Gamma \to \mathbb{Z}^m \to 1
$$
then must we have
$$
\pi_1(M) \cong \Gamma
$$
for some $ n+m $ dimensional compact solvmanifold $ M $?
A similar question: Is every torus bundle over a torus a solvmanifold? In other words, if we have a fiber bundle
$$
T^n \to M \to T^m
$$
then can we conclude that $ M $ is a solvmanifold?
This is true for $ n=m=1 $. The only extension of $ \mathbb{Z} $ by $ \mathbb{Z} $ are $ \mathbb{Z}^2 $ and the Klein bottle group. And indeed the compact 2d solvmanifolds are exactly $ T^2 $ and the Klein bottle $ K $. These two are also the only circle bundles over the circle.
For dimension 3 I'm a bit less sure but I think all the bundles
$$
T^1 \to M \to T^2
$$
are solvmanifolds. And I know that many of the bundles
$$
T^2 \to M \to T^1
$$
are solvmanifolds.
Best Answer
The answer, comments and references from Igor Belegradek here
https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds
prove that something much stronger is true: A manifold $ M $ is the total space of a bundle $$ N \to M \to T^n $$ where $ N $ is a compact nilmanifold and $ T^n $ is a torus if and only if $ M $ is homeomorphic to a compact solvmanifold.
His answer also shows that $ \Gamma $ fits into a SES $$ 1 \to N \to \Gamma \to \mathbb{Z}^m \to 1 $$ where $ N $ is nilpotent torsion free and finitely generated if and only if $ \Gamma $ is the fundamental group of a compact solvmanifold. Since compact solvmanifolds are determined up to diffeomorphism by their fundamental group this shows that the answer to the first question is yes as well.
The smooth case is also addressed. Corollary 2.3 of https://arxiv.org/pdf/1307.3223.pdf states that "If $ h: T^n → T^n $ , $ n ≥ 6 $, is a diffeomorphism given by Proposition 2.1 then the mapping torus $ M_h $ is a fake torus." By fake torus the authors mean $ M_h $ is homeomorphic to a torus but not PL homeomorphic. Since $ M_h $ is not PL homeomorphic to a torus it is also not diffeomorphic to a torus. Thus already in dimension 7 we have an exotic homotopy torus $ M_h $ which is the total space of a torus bundle over the torus $$ T^6 \to M_h \to S^1 $$ but which is not diffeomorphic to any solvmanifold. So in every dimension $ d \geq 7 $ there are smooth torus bundles over a torus (both base and fiber with the standard smooth structure) that are not diffeomorphic to any solvmanifold.
For $ d \leq 3 $ we have by Moise's theorem that there are no exotic smooth structures. So all torus bundles over a torus with total space of dimension $ d \leq 3 $ are diffeomorphic to a solvmanifold.
For $ d=4 $ the existence of exotic 4 tori (or any exotic closed aspherical manifold) seems to be an open problem see
https://mathoverflow.net/questions/133797/do-there-exist-exotic-4-tori
In that same question it is pointed out that there are exotic tori in $ d=5,6 $ but it is not clear that these can be constructed as smooth fiber bundles with base and fiber standard tori.
That said, for the $ d=4 $ case a torus bundle over a torus has base and fiber with dimension $ \leq 3 $ so there shouldn't be any exotic gluing maps and it seems like the total space of such a bundle should just always have the standard smooth structure.
For $ d=5,6 $ I have no idea what kind of smooth structures might or might not be possible on torus bundles over the torus.