A group \(G\) is called Jordan if there exists a positive integer \(J\) such that every finite subgroup \(B\) of \(G\) contains an normal abelian subgroup \(A\) of \(B\) of index \([B:A]\leq J\). The group \(G\) is strongly Jordan if it is Jordan and its finite abelian subgroups are generated by at most \(a=a(G)\) elements. In a paper under review the authors study the group \(\text{Aut}(W)\) of (biregular) automorphisms of an irreducible algebraic variety \(W\) over an algebraically closed field \(k\) of characteristic zero. The main result is the following. Let \(W\) be an irreducible quasiprojective variety that is birational to a product \(A\times{\mathbb P}^1\), where \(A\) is a smooth projective variety which does not contain rational curves. Then \(\text{Aut}(W)\) is strongly Jordan. As a corollary the authors obtain that if \(W\) is a quasiprojective irreducible variety of dimension \(d\leq 3\) which is not birational to \(E\times{\mathbb P}^2\), where \(E\) is an elliptic curve, then \(\text{Aut}(W)\) is Jordan.