A general problem is to understand all (injective) homomorphisms between (finite index subgroups of) mapping class groups of surfaces. Birman and Hilden proved that if $S\rightarrow X$ is a regular branched covering space of surfaces, there is an embedding of the subgroup of the mapping class group of $X$ consisting of mapping classes that have representatives that lift to $S$ in the mapping class group of $S$ modulo the group of deck transformations. This relationship does not always hold for irregular branched covers. We give a necessary condition and a sufficient condition for when such an embedding exists. We also give new explicit examples that satisfy the necessary condition and examples that do not satisfy the sufficient condition.