Is semi-direct product converted to direct product if the normal subgroup is the center of $G$?

Question

Suppose $G$ is a group and $N$ is a normal subgroup in $G$. Also suppose $G=N \rtimes H$. I need to know, is this semi-direct product reduced to the direct product if $N=Z(G)$?

My initial guess is yes! But any detailed explanation will be highly appreciated.