If $N\in \mathbb N$ is a palindrome in base $b$, can we say it is a palindrome in other bases?

Question

I have noticed a pattern among palindromic numbers, and was curious if there is a general truth out there. My examples are limited to smaller bases so that numerals may be used. My conjecture is that if $N_b$ is a palindrome with $b=p^n$ for some prime $p$ and all digits of $N_b$ are less than $p$ then $N_p$ is also a palindrome. For example,

$$10100101_8 = 1000001000000001000001_2$$ $$12022021_9 = 102000202000201_3$$

Using only divisibility as a criterion, the conjecture does not hold, since the first example above is equal to $20020001001_4$.

My central question is pretty big I guess. If $N_b$ is a palindrome, for which $a$ is $N_a$ a palindrome? I'm interested in any subcases such as this one as well.

Answer 1

Counter-example (for part 1) : $$45611211654_{49} = 405060101020101060551_{7}$$