Conjugacy-closedness is not upper join-closed

Statement
We can have the following situation: $$H \le G$$ is a subgroup, $$K_1, K_2$$ are intermediate subgroups of $$G$$ containing $$H$$, such that $$H$$ is conjugacy-closed in $$K_1$$ as well as in $$K_2$$, but not in the join $$\langle K_1, K_2$$.

A generic example
Suppose $$H$$ is a group with an automorphism $$\sigma$$ of order two that is not class-preserving: the automorphism does not preserve conjugacy classes. For instance, $$H$$ could be an Abelian group of exponent greater than two, and $$\sigma$$ could be the inverse map.

Let $$A$$ be the subgroup of $$\operatorname{Aut}(H \times H)$$ generated by the automorphism $$(\sigma,\sigma)$$ (i.e., $$\sigma$$ acting coordinate-wise) and the coordinate exchange automorphism. Since both these automorphisms are of order two and commute, $$A$$ is a Klein-four group. Define $$G = (H \times H) \rtimes A$$ with the specified action.

Now, let $$B_1$$ be the two-element subgroup of $$A$$ generated by the coordinate exchange automorphism, and $$B_2$$ be the two-element subgroup of $$A$$ generated by the composite the coordinate exchange automorphism and $$(\sigma,\sigma)$$. Define $$K_1 = (H \times H)B_1$$ and $$K_2 = (H \times H)B_2$$.


 * The subgroup $$H = H \times \{ e \}$$ is conjugacy-closed in $$K_1$$: Any element of $$K_1$$ is a product of an element in $$H \times H$$ and an element in $$H \times H$$. Note that the element of $$H \times H$$ preserves conjugacy classes, so it remains to see the effect of the element of $$B_1$$. if the element of $$B_1$$ is trivial, it of course acts as the identity. If it is not trivial, it sends every element of $$H \times \{ e \}$$ to an element of $$\{ e \} \times H$$, so no two distinct conjugacy classes in $$H \times \{ e \}$$ get fused by the action of $$B_1$$.
 * The subgroup $$H = H \times \{ e \}$$ is conjugacy-closed in $$K_1$$: The reasoning is identical to the above reasoning.
 * The subgroup $$H = H \times \{ e \}$$ is not conjugacy-closed in $$G$$: Indeed, the automorphism $$(\sigma,\sigma)$$ does not preserve conjugacy classes in $$H \times \{ e \}$$.

A similar kind of example can be constructed when the automorphism $$\sigma$$ is not class-preserving, and has finite order $$n > 2$$. In this case, we need to case a $$n$$-fold direct product of $$H$$, and have the diagonal automorphism $$\sigma$$ as well as the cyclic coordinate permutation automorphism act on this direct product.

Some particular examples
The smallest particular example of the above is when $$H$$ is the cyclic group of order three, and $$\sigma$$ is the inverse map. $$G$$ in this case has order $$36$$, and $$K_1, K_2$$ both have order $$18$$.

Within nilpotent groups, the smallest particular example is when $$H$$ is the cyclic group of order four, and $$\sigma$$ is the inverse map. $$G$$ has order $$64$$, and $$K_1, K_2$$ both have order $$32$$.