Suppose $f:M\rightarrow N$ is a submersion. Then, $M\times_NM$ is a smooth manifold from Transversal theorem.
Suppose $\theta:G\rightarrow H$ is a morphism of Lie groups. Assume that it is a submersion. Does it imply $G\times_H G$ is a Lie group?
As $G\rightarrow H$ is submersion, $G\times_H G $ is a smooth manifold. Does it have to be Lie group?
When will $G\times_H G$ is a Lie group?
Yes it’s always a Lie group.
Closure under multiplication and inverses you can check, so it’s a group. For smoothness, note $M\times_NM$ isn’t just a manifold, but an embedded submanifold of $M\times M$. The multiplication and inversion for $G\times_HG$ are restricted from $G\times G$ to the submanifold, hence smooth.