Rank growth of elliptic curves in $S_4$ and $A_4$ quartic extensions of the rationals

We investigate the rank growth of elliptic curves from $\mathbb{Q}$ to $S_4$ and $A_4$ quartic extensions $K\mathrm{/}\mathbb{Q}$. In particular, we are interested in the quantity $\mathrm{r}\mathrm{k}(E\mathrm{/}K)-\mathrm{r}\mathrm{k}(E\mathrm{/}\mathbb{Q})$ for fixed $E$ and varying $K$. When $\mathrm{r}\mathrm{k}(E\mathrm{/}\mathbb{Q})\le 1$, with $E$ subject to some other conditions, we prove there are infinitely many $S_4$ quartic extensions $K\mathrm{/}\mathbb{Q}$ over which $E$ does not gain rank, i.e. such that $\mathrm{r}\mathrm{k}(E\mathrm{/}K)-\mathrm{r}\mathrm{k}(E\mathrm{/}\mathbb{Q})=0$. To do so, we show how to control the 2-Selmer rank of $E$ in certain quadratic extensions, which in turn contributes to controlling the rank in families of $S_4$ and $A_4$ quartic extensions of $\mathbb{Q}$.