Let's have an elliptic curve $E$ defined as $y^2=x^3+b$ in a finite field with prime characteristic $p$ and its sextic twist $E_6$. On both curves, let's choose a "generator" point $P$ (resp. $P_6$) such that x-coordinate of $P$ is equal to x-coordinate of $P_6$.
Additionally, let's have another point $Q$ which is an i-th scalar multiple of P, i.e. $Q=P*i$.
Is there a way how one could use one's knowledge of $Q$ in order to do a sort of "point-wise" jumps between $E$ and $E_6$, i.e. identify a "corresponding" $Q_6$ such that $Q_6=P_6*i$ without knowing $i$ ?
Somehow my intuition tells me that this should potentially be possible given that $E$ a $E_6$ are supposed to be isomorph in $F_{p^2}$ but given my lack of proper higher math education - e.g. I have no clue what is a difference between "isomorph" and "isogenous" and only recently discovered motivation behind the notion of field extension - it may also be the case that I am completely wrong.
(Also please accept my apologies if this question is too trivial, using incorrect terminology or not properly formatted [my first post])
Thanks in advance for any kind of hint.