# Semidirect product of metaplectic group and Heisenberg group

I know that Symplectic group has an action on Heisenberg group.

I am wondering how to extend this to non-trivial two fold metaplectic covering?

I presume you are looking for a faithful action of $$Mp_{2n}$$ on something related to the Heisenberg group $$H_{2n+1}$$. This is well-known as Weil Representation.
In the modern language, consider $$Mp_{2n}$$ acting on $$H_{2n+1}$$ by automorphisms. This action has a kernel. Now consider the action on the category of unitary representations of $$H_{2n+1}$$ by twisting representations by automorphisms. This categorical representation still has the same kernel. Finally, choose a skeleton of the category of unitary representations of $$H_{2n+1}$$. The modern interpretation of all this Weil and Stone–von-Neumann business is that $$Mp_{2n}$$ acts on the skeleton and this yields a faithful categorical representation of $$Mp_{2n}$$.