A wreath product of semigroups and varieties of finite index

We have found the necessary and sufficient conditions on semigroups under which their extended standard wreath product generates a variety of finite index. Further, the similar problem is solved for varieties of completely simple semigroups. Finally, the necessary and sufficient conditions on monoid $S$ are found under which the endomorphism monoid of a free left act over $S$ generates a variety of finite index.