Permutably complemented does not satisfy transfer condition

Statement
It can happen that $$H, K \le G$$ are subgroups such that $$H$$ is permutably complemented in $$G$$ but $$H \cap K$$ is not a permutably complemented subgroup in $$K$$.

Example of the dihedral group
Let $$G$$ be the dihedral group of order eight:

$$G = \langle a, x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

Consider the two subgroups:

$$H = \langle a^2, x \rangle = \{ e, a^2, x, a^2x \}, \qquad K = \langle a \rangle = \{ e,a,a^2,a^3 \}$$.

Then, we have:


 * $$H$$ is a permutably complemented subgroup of $$G$$: The subgroup $$\{ e, ax \}$$ is a permutable complement to $$H$$ in $$G$$.
 * $$H \cap K$$ is not a permutably complemented subgroup of $$G$$: $$H \cap K$$ is the group $$\{e, a^2 \}$$, which clearly does not have any permutable complement inside $$K$$, which is a cyclic group of order four.