Galois groups in a family of dynatomic polynomials

For every nonconstant polynomial $f\in\mathbb Q[x]$, let $\Phi_{4,f}$ denote the fourth dynatomic polynomial of $f$. We determine here the structure of the Galois group and the degrees of the irreducible factors of $\Phi_{4,f}$ for every quadratic polynomial $f$. As an application we prove new results related to a uniform boundedness conjecture of Morton and Silverman. In particular we show that if $f$ is a quadratic polynomial, then, for more than $39\%$ of all primes $p$, $f$ does not have a point of period four in $\mathbb Q_p$.