Is it true that the mapping torus of a solvmanifold is always a solvmanifold?
Some relevant facts, many from Flat 3 manifolds and mapping tori of flat surfaces :
Every mapping torus of a compact nilmanifold is a (compact) solvmanifold see Torus bundles and compact solvmanifolds
The 1d solvmanifolds are $ S^1, \mathbb{R} $. The mapping tori of $ S^1 $ are $ T^2,K^2 $ and the mapping tori of $ \mathbb{R} $ are $ \mathbb{R} \times S^1, M $ where $ M $ is the Mobius strip. These mapping tori are all solvmanifolds.
The 2d solvmanifolds are $ \mathbb{R}^2, T^2,K^2,\mathbb{R} \times S^1, M $.
The mapping tori of $ T^2 $ are given here Bianchi classification of solvable Lie groups and cocompact subgroups or here https://mathoverflow.net/questions/414531/3-dimensional-solvmanifolds-and-thurston-geometries/416509#416509 in particular they are given by the trace of the mapping class and whether it is diagonalizable. Also they are all solvmanifolds since $ T^2 $ is a compact nilmanifold.
The mapping tori of $ \mathbb{R}^2 $ are $ \mathbb{R}^2 \times S^1, \mathbb{R} \times M $. They are both solvmanifolds.
The mapping tori of the cylinder $ S^1 \times M , \mathbb{R} \times T^2, \mathbb{R} \times K^2 , X $. The first three are solvmanifolds, unsure about the $ X $.
The mapping tori of the Mobius strip $ M $ are $ S^1 \times M, Y $. The first is a solvmanifold, unsure about $ Y $.
The four mapping tori of $ K^2 $ are given here Mapping Torus of Klein bottle they are exactly the four flat compact nonorientable 3 manifolds. The trivial product $ S^1 \times K^2 $ is a solvmanifold. Unsure about the other three