When I was computing the conjugacy classes of $A_4$, I found that $\{(12)(34), (13)(24),(14)(23)\}$ is one of the conjugacy classes of $A_4$. This is not hard to see as the center of $A_4$ is trivial. However, we have that
$(23) * (12)(34) * (23) = (13)(24)$
How is this possible? How $(12)(34)$ and $(13)(24)$ can be conjugates if $(23) \not \in A_4$?
It is possible as the following conjugation relation also holds $$(132)\circ (12)(34)\circ (123)=(13)(24)$$
In other words, your two Klein-type (I am referring to the Klein group) elements are conjugates within $A_4$, so there is nothing to worry about. In general, if $G$ is an arbitrary group, $H \leqslant G$ a subgroup and $a, b \in H$ are conjugates in $H$ that doesn't preclude the possibility of $a$ being conjugated into $b$ by an element in $G\setminus H$.