Let us fix the base field $k=\mathbb Q$. Then, say on the etale site, $\mu_p$ is a $\mathbb Z/p$ torsor since locally (after base change to $Q(\mu_p)$), we can pick a root of unity, aka a section after which the two sheafs are isomorphic.
This implies that there should be a map (over $\mathbb Q$) $f: \mathbb Z/p\times \mu_p \to \mu_p$ that is locally just the multiplication map $m: \mathbb Z/p\times \mathbb Z/p \to \mathbb Z/p$ for any choice of section of $\mu_p$.
How do we define $f$ explicitly?
There is no such action (and consequently, $\mu_p$ is not a torsor under $\mathbb{Z}/p$), unless $k$ has a primitive root of unity (and so $\mu_p \cong \mathbb{Z}/p$). Indeed, if such an action would exist over $k$, then the image of $$ (1,0) \in \mathbb{Z}/p \times \mu_p $$ would be a primitive root of unity defined over $k$.