If $H$ and $G/H$ are path connected, then is $G$ path connected?

Question

Let $G$ a topological group and $H$ a subgroup such that $H$ and $G/H$ are path connected.
Is it true this implies that $G$ is path connected?

I already know that if $H$ and $G/H$ are connected so is G (proposition 1.6.5), but i can't find anything about path connected spaces.