Summary: A subgroup \(X\) of a group \(G\) is said to be \(\textit{pronormal}\) if for each element \(g\) of \(G\) the subgroups \(X\) and \(X^g\) are conjugate in \(\langle X,X^g\rangle\). The aim of this paper is to study pronormality and some close embedding properties, like weak normality and weak pronormality. In particular, it is proved that these properties can be countably detected, and the behaviour of groups which are rich in (generalized) pronormal subgroups is investigated.