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\begin{document}$${\mathfrak {X}}$$\end{document} be a class of finite groups closed under subgroups, homomorphic images, and extensions. We study the question which goes back to the lectures of H. Wielandt in 1963–1964: Given an X\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {X}}$$\end{document}-subgroup K and a maximal X\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {X}}$$\end{document}-subgroup H, is it possible to detect embeddability of K in H (up to conjugacy) by their projections into the factors of a fixed subnormal series? On the one hand, we construct examples where K has the same projections as some subgroup of H but is not conjugate to any subgroup of H. On the other hand, we prove that if K normalizes the projections of a subgroup H, then K is conjugate to a subgroup of H even in the more general case when H is a submaximal X\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {X}}$$\end{document}-subgroup.
