On the Order of the Schur Multiplier of a Pair of Finite p-Groups II

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Let G be a finite p-group and N be a normal subgroup of G, with |N|=pⁿ and |G/N|=pᵐ. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair (G,N) of finite p-groups is bounded by p¹/²ⁿ⁽²ᵐ⁺ⁿ⁻¹⁾ and hence it is equal to p¹/²ⁿ⁽²ᵐ⁺ⁿ⁻¹⁾⁻ᵗ, for some non-negative integer t. Recently the authors characterized the structure of (G,N) when N has a complement in G and t≤ 3. This paper is devoted to classify the structure of (G,N) when N has a normal complement in G and t=4,5.