I have come across an exercise asking for a proof of something that is definitely false: If $G$ is a Lie group, $H$ a connected closed subgroup and $G/H$ simply connected, then $G$ is itself simply connected. A counter example is $SO(3)/SO(2) \cong \mathbb{S}^2$.
However, I have been trying to figure out if this statement is actually true if $H$ is simply connected rather than just connected. I have tried to prove it by patching a homotopy from $G/H$ together and lifting it to $G$, but it does not quite work - I can not get the endpoints to match, even when using the simple connectedness of $H$.
I have not yet found any counter examples to this modified statement. I am not asking for a proof, but rather, is the statement true or false for $H$ simply connected?