Let $A$ be a group with subgroup $H=\langle h \rangle \times C$, where $h$ is infinite order and $C$ is finite central in $A$. Suppose there exists $M \lhd_{f} A$ such that $M \cap \langle h \rangle=\langle h^{r} \rangle$ where $r$ is any positive integer. It is clear that $\langle h \rangle \cap C=1$ and $h, h^{2}, \ldots, h^{r-1} \notin M$.
Question 1: are $h, h^{2}, \ldots, h^{r-1} \notin MC$? if not, is there any condition(s) can be added to make it true?
Question 2: if there's an additional condition, $c, c^{2}, \ldots, c^{r-1} \notin M$, in what condition we can have $hc, h^{2}c^{2}, \ldots, h^{r-1}c^{r-1} \notin M$ or $hc, h^{2}c^{2}, \ldots, h^{r-1}c^{r-1} \notin MC$?