connection between integrablity on the locally compact group and compact subgroup of it

Question

Let $G$ be an locally compact group with Haar measure $dx$ and $H$ is compact subgroup of it with normalize Haar measure $dh$. If $F$ belong to $L^1(G)$, the restriction of $F$ to $H$ belong to $L^1(H)$ (i.e. $F$ is integrable respect to $dh$ on $H$)? Conversely, can we extend every member of $L^1(H)$ to some member of $L^1(G)$?