My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that preserve the fiber structure of the projection $\mathbb T\times\mathbb R\to\mathbb T$ in the sense that $h(\{z\}\times\mathbb R)=\{h(z)\}\times\mathbb R$ for any $z\in\mathbb T$ where $\mathbb T:=\{z\in \mathbb C:|z|=1\}$ is the unit circle on the complex plane.

As I understand, the group $G$ is a semidirect product $H(\mathbb T)\rtimes C(\mathbb T,H(\mathbb R))$ of the homeomorphism group $H(\mathbb T)$ of the circle and the group $C(\mathbb T,H(\mathbb T))$ of continuous maps form the circle to the homeomorphism group $H(\mathbb R)$ of the real line.

Do you know any paper that studies this automorphism group $G$ (from algebraic or topological point of view)? Maybe in a more general context of automorphism groups of fiber bundles or foliations?