Bounds for the index of the second center subgroup of a pair of finite groups

By a pair of groups, we mean a group G and a normal subgroup N. ##In the present work, we give an upper bound for |N/Z2(G,N)| in which Z2(G,N) denote the second center subgroup of a pair (G,N) of finite groups where N is a subgroup of G. ##As a consequence, we obtain an upper bound for K/Z(H,K) where H ≅G/Z(G,N) and K ≅ N/Z(G,N), for a pair (G,N) of finite groups