Let \(G\) be a finite group and \(\Sigma\): \(G_0\leq G_1\leq\cdots\leq G_n\) a series of subgroups of \(G\). A subgroup \(A\) of \(G\) is said to be \(\Sigma\)-embedded in \(G\) if for each pair \((K,H)\), where \(K\) is a maximal subgroup of \(H\) and \(G_{i-1}\leq K<H\leq G_i\) for some \(i\), either \(A\cap H=A\cap K\) or \(AH=AK\). A subgroup \(A\) of \(G\) is said to be nearly \(m\)-embedded (\(m\)-embedded) in \(G\) if \(G\) has a subgroup (subnormal subgroup) \(T\) and a \(\{1\leq G\}\)-embedded subgroup \(C\) in \(G\) such that \(G=AT\) and \(T\cap A\leq C\leq A\).
Here are some of the major results. If every minimal subgroup of \(G\) is nearly \(m\)-embedded in \(G\), then \(G\) is \(2'\)-supersoluble (Theorem 3.1). If \(p\) is a prime dividing \(|G|\) such that \(\gcd(|G|,p-1)=1\), then \(G\) is \(p\)-nilpotent exactly if either the Sylow \(p\)-subgroups of \(G\) have order \(p\) or there is an integer \(k\) such that \(1\leq k<n\) and every subgroup of \(G\) of order \(p^k\) and every subgroup of order 4 (if \(p^k=2\) and \(P\) is non-Abelian) is \(m\)-embedded in \(G\) (Theorem 4.1). If \(\mathcal F\) is a saturated formation containing all supersoluble groups and \(G\) has a normal subgroup \(E\) such that \(G/E\in\mathcal F\) and every Sylow subgroup \(P\) of \(E\) has the property that either every maximal subgroup of \(P\) or every cyclic subgroup of \(P\) of prime order and of order 4 (if \(P\) is a non-Abelian 2-group) is nearly \(m\)-embedded in \(G\), then \(G\in\mathcal F\) (Theorem 5.1). If \(\mathcal F\) is a saturated formation containing all supersoluble groups and \(G\) has a normal subgroup \(E\) such that \(G/E\in\mathcal F\) and every maximal subgroup of every Sylow subgroup of \(F^*(E)\) is nearly \(m\)-embedded in \(G\), then \(G\in\mathcal F\) (Theorem 5.2). If \(G\) is soluble, then \(G\) is supersoluble if and only if every subnormal subgroup of \(G\) is \(\{1\leq G\}\)-embedded in \(G\) (Theorem 6.2).