Summary: A chief factor \(H/K\) of a group \(G\) is called \(\mathcal F\)-central if \([H/K](G/C_G(H/K))\in\mathcal F\). The product of all such normal subgroups of \(G\), whose \(G\)-chief factors are \(\mathcal F\)-central in \(G\), is called the \(\mathcal F\)-hypercentre of \(G\) and denoted by \(Z_{\mathcal F}(G)\). The finite groups \(G\) with factorizations \(G=AB\), where \(A\cap B\leq Z_{\mathcal F}(G)\) for some class of groups \(\mathcal F\), are studied. Some known results about factorizations of groups are generalized.