Sylow does not satisfy transfer condition

Statement
It is possible to have a finite group $$G$$, a Sylow subgroup $$H$$, and a subgroup $$K$$ of $$G$$ such that $$H \cap K$$ is not a Sylow subgroup of $$K$$.

Related facts

 * Hall does not satisfy transfer condition
 * Normal Sylow satisfies transfer condition
 * Normal Hall satisfies transfer condition

Property-theoretic proof
We know that the property of being a Sylow subgroup is transitive (a Sylow subgroup of a Sylow subgroup is Sylow). Thus, if the property of being Sylow satisfies the transfer condition, we have that the property of being a Sylow subgroup is intersection-closed, by the general fact Transitive and transfer condition implies finite-intersection-closed.

On the other hand, an intersection of Sylow subgroups need not be Sylow.