Is there an expression for the Riemannian holonomy of a quotient of Lie groups (not necessarily compact or simply connected) such that $G/H$ is simply connected? In particular, is there a general way to write $Hol(G/H, g_{G/H})$ in terms of $Hol(G, g_{G})$ and $Hol(H, g_{H})?$ Moreover, for what $G$ and $H$ does $Hol(G/H)=Hol(G)/Hol(H)$ or $Hol(G/H)=G/H$?

I have been stumbling on this problem for a while, and I cannot seem to figure it out. I would appreciate any help. Thanks!