Order-conjugate and Hall not implies order-dominated

Statement
It is possible to have a finite group $$G$$, an order-conjugate Hall subgroup $$H$$ of $$G$$, and a subgroup $$K$$ of $$G$$ whose order is a multiple of the order of $$H$$ such that no conjugate of $$H$$ is contained in $$K$$.

Proof
Consider the projective special linear group $$G = PSL(2,61)$$. Let $$H$$ be a $$\{2,5 \}$$-Hall subgroup of $$G$$ (it turns out that such a $$H$$ is a dihedral group of order twenty) and $$K$$ be a subgroup of $$G$$ isomorphic to the alternating group of degree five. Then, we have the following:


 * $$H$$ is conjugate to any other subgroup of the same order:
 * The order of $$H$$ divides the order of $$K$$: Indeed, $$H$$ has order $$20$$ and $$K$$ has order $$60$$.
 * No conjugate of $$H$$ is contained in $$K$$: In fact, the alternating group of degree five contains no subgroups of order $$20$$.