I was looking at the primitive roots $n \bmod p$ and $p^2$ to see how often we get primitive roots of a prime that are not primitive roots of the square of that prime. I'll call this a reluctant root of $p$.
It's really rare.
With $p$ limited to a little under $38000$, I didn't find any such cases at all for $n<10$. In particular I started out looking specifically for cases when $2$ is a reluctant root, and finding none I widened the search. The first case I found was $n=11, p=71$, when I was just looking at $n$ prime, but there is also a case for $n=10, p=487$. In these cases, as in most others, and taking only $n<p,$ this was the only such case despite many hundreds of primes for which $n$ was a primitive root.
Searching the other way around, looking for reluctant roots of various primes $p,$ I found that in the first $400$ primes, which have a total of $190111$ primitive roots, there are only $143$ reluctant roots across $122$ primes. Most primes have no reluctant roots; the first few primes that have reluctant roots are $29, 37, 43, 71, 103, 109, 113, 131\ldots$; even fewer ($18$ in this range) have more than $1$, with $653$ taking the golden cupcake (so far) with $4$ reluctant roots - $84, 120, 287, 410$.
Main question: what is the smallest prime that has $2$ as a reluctant root, or is there no such prime?
Other questions: why are reluctant roots so rare? and is there any more investigative work on these (perhaps under a less Harry-Potteresque name)?