# The automorphism group of the fibered cylinder

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?