The paper deals with the inverse differential Galois problem for algebraic groups having semisimple identity component. The authors give a criterion for a differential equation to have a given semisimple algebraic group as a Galois group. Using this criterion for $H$-equivariant differential equations, they deduce a procedure to construct differential equations with Galois group $G$, where $G= H\ltimes G^0$, $H$ is finite and $G^0=G_1\cdots G_r$ with each $G_i$ of type $A_{l},C_{l},D_{l},E_6$ or $E_7$.
