I assume you are asking about $N(m)$, the number of distinct integers $n$ which satisfy $\phi(n)=m$ where $\phi$ is the Euler Totient function.
It turns out that this function is complicated, and there
There are many papers providing both results regarding upper and lower bounds for the size of $N(m)$, as well as the mean and variance. Take a look at Carl Pomerance's In particular, Carmichael conjectured that $N(m)$ is never equal to $1$.
Pomerance gave the upper bound $$N(m)\leq m\exp{-(1+o(1))\log m \frac{\log \log \log m}{\log \log m}}$$ and also showed that there are infinitely many $m$ for which $$N(m)\geq m^{\frac{5}{9}}.$$
Bateman showed that $$\sum_{m\leq x} N(m)=\frac{\zeta(2)\zeta(3)}{\zeta(6)}x+O\left(xe^{-c\sqrt{\log x\log \log x}}\right),$$ and we also have that $$\sum_{m\leq x} N(m)-\frac{\zeta(2)\zeta(3)}{\zeta(6)}x=\Omega\left(x^\frac{5}{9}\right)$$
For more details, see the following paper of Pomerance: Popular Values of Euler's Function.
