Solutions to $\varphi(\varphi(a))=\varphi(\varphi(b))$ As the title indicates, I'm looking for what is known about solutions to $\varphi(\varphi(a))=\varphi(\varphi(b))$ [other than "trivial cases" for which  $\varphi(a)=\varphi(b),$] where $\varphi$ is the totient function, i.e. at $n$ it is the number of positive integers at most $n$ and coprime to it.
I became interested in the question while looking at primitive roots, for which it's known that, if $n$ has any primitive roots at all, then there are $\varphi(\varphi(n))$ of them. I did find the primes $11,13$ each have $4$ primitive roots, and began to wonder if there were more examples. However due to my lack of expertise/software, I decided to ignore the assumption about $a,b$ in my question having primitive roots. Then I could find more examples, and besides that I thought such a question about the totient composed with itself might be of interest, or maybe had been investigated already somewhere. Any info appreciated.
Edit--I did a search using  table for primes (and a few prime powers) below $100$ and found several examples. (Still interested in the general situation but now more in the odd prime power case.) 

Not an answer, just a visualization:
import math
import matplotlib.pyplot as plt

def phi(x):
    result = []
    for n in x:
        amount = 0
        for k in range(1, n + 1):
            if math.gcd(n, k) == 1:
                amount += 1
        result.append(amount)
    return result

x = list(range(2000))

fig,axes = plt.subplots()
plt.scatter(x,phi(phi(x)),s=1)
fig.suptitle('phi(phi(x))')
plt.show()


compare to phi(x):

