I have a normal distribution $N(0, \sigma^2)$. If I start selecting all distributions $N(\mu, \sigma^2/2)$ from this distribution, what are the parameters of the distribution for the mean $\mu$: $N(0, ?)$? What will the standard deviation be?

I have a normal distribution $N(0, \sigma^2)$. If I start selecting all distributions $N(\mu, \sigma^2/2)$ from this distribution, what are the parameters of the distribution for the mean $\mu$: $N(0, ?)$? What will the standard deviation be?

In other words, if there are a set of $N(\mu, \sigma^2/2)$ distributions that together form $N(0, \sigma^2)$ distribution, what the distribution for $\mu$ should be? There are two extreme points. This is if I chose distributions with the same standard deviation $\sigma^2$. In this case, the standard deviation for $\mu$ would tend to zero with n->inf. And if I were to select one point from $N(0, \sigma^2)$, then the standard deviation for $\mu$ would be equal to $\sigma^2$.