Impulse response of an infinite-length system
Cconsider the following pair of signals and their transforms:
If $g[n]$ has a Z-transform $G(z)$ then the expanded (zero stuffed) signal $h[n]$ has the Z-transform $H(z) = G(z^N)$, where
$$ h[n] = \begin{cases} {g[\frac{n}{N}] ~~~,~~~ n = m N  \\ 0 ~~~~~~~~~,~~~ \text{otherwise}  \tag{1}}\end{cases} $$
Looking at the given Z-transform :
$$H(z) = \frac{ 1}{ 1 - \alpha z^{-N} }$$
it can be seen that if $H(z) = G(z^N)$ where
$$G(z)= \frac{ 1}{ 1 - \alpha z^{-1} }$$
and $g[n] = \alpha^n u[n] $.
Then you can simply conlcude that :
$$ h[n] = \alpha^{n/N} ~u[n/N] =  \begin{cases} {\alpha^{n/N}  ~~ ,~~ n = m N, m = 0,1,..  \\ 0 ~~~~~~~~~,~~ \text{otherwise} }\end{cases} $$