Left cosets of a subgroup are not in bijection via left multiplication in a monoid

Statement
It is possible to have a monoid $$G$$ and a subgroup $$H$$ of $$G$$ such that the left cosets of $$H$$ (i.e., sets of the form $$gH$$) in $$G$$ are not in bijection via left multiplication.

Related facts

 * Left cosets of a subgroup partition a monoid
 * Left cosets of a subgroup are in bijection via left multiplication in a cancellative monoid

Proof
Consider $$G$$ to be the multiplicative monoid of the real numbers and $$H$$ to be the subgroup of nonzero real numbers. Then, $$H$$ has two left cosets in $$G$$: $$H$$ itself and the one-element set $$\{ 0 \}$$. Clearly, there is no bijection via left multiplication between $$H$$ and $$\{ 0 \}$$.