I recently asked a question regarding the equality for two apparently different definitions for the projective special unitary group. (Equality between two definitions of the projective special unitary group). The problem was to show $$\frac{(GU(3,q) \cap SL(3,q^2))Z}{Z} = \frac{GU(3,q)Z}{Z} \cap \frac{SL(3,q^2)Z}{Z}.$$ Containment from the left to the right is easy to see, but the proof of the other direction was surprisingly nontrivial.
This situation leads me to ask if there are some known conditions on subgroups $H,K,N$ of a group $G$ with $N\trianglelefteq G$ such that $$\frac{(H\cap K)N}{N} = \frac{HN}{N} \cap \frac{KN}{N}.$$ Is this something that holds in general? if not, which conditions are necessary and/or sufficient for this to be the case?