Let $F$ be a field of characteristic not equal to $2$.

Assume $L/F$ is a field extension of degree $6$ with no intermediate subfields.

We know by a result due to Joubert (1867), later improved by H. Kraft (2006), that one can find $\alpha$ $\in L$ such that $L = F(\alpha)$ and $min_{\alpha , F}(X)$ = $X^6 + a_4X^4 + a_2X^2 + a_1X^1 + a_0$ where $a_i \in F^{*}$ and $a_1 = a_0 \neq 0$. This result concerns general separable field extensions of degree $6$ in characteristic $\neq 2$. My questions is whether this can be improved under the additional assumption that the Galois group corresponding to the normal closure of $L$ is either $S_6$ or $A_6$.

In particular, I am trying to reduce to the case where $min_{\alpha , F}(X)$ = $X^6 + a_1X^1 + a_0$ or $min_{\alpha , F}(X)$ = $X^6 + a_5X^5 + a_0$.

Any references to the literature outside of that mentioned above is appreciated!