Groups satisfying a strong complement property

Let G=NH be a finite group where N is normal in G and H is a complement of N in G. For a given generating sequence (h1,…,hd) of H we say that (N,(h1,…,hd)) satisfies the strong complement property, if 〈h1xjavax.xml.bind.JAXBElement@7f1ba15f,…,hdxjavax.xml.bind.JAXBElement@4fccb86〉 is a complement of N in G for all x1,…,xd∈N. When d is the minimal number of elements needed to generate H, and (N,(h1,…,hd〉)) satisfies the strong complement property for every generating sequence (h1,…,hd) with length d, then we say that (N,H) satisfies the strong complement property. In the case when |N| and |H| are coprime, we show that (N,H) can only satisfy the strong complement property if H is cyclic or if H acts trivially on N. We give on the other hand a number of examples that show this does not need to be the case when considering the strong complement property of (N,(h1,…,hd)) for a given fixed generating sequence. In the case when N and H are not of coprime order, we give examples where (N,H) satisfies the strong complement property and where H is not cyclic and does not act trivially on N.