I assumed it was true, and then found
$f_Y(y) = f_x(\frac{y}{a}).a^{-1} = a^{-1}\frac{1}{\sqrt{2\pi\sigma^2}}.\exp\{-\frac{(\frac{y}{a}-\mu)}{2\sigma^2}\} = \frac{1}{\sqrt{a^2.2\pi\sigma^2}}.\exp\{\frac{(y-\frac{\mu}{a})}{2a\sigma^2}\} $
which is almost of the form of a normal pdf, but there isn't a consistent value for $\sigma^2_Y$, unless $a=\pm1$