  We investigate the rank growth of elliptic curves from $\mathbb{Q}$ to $S_4$
and $A_4$ quartic extensions $K/\mathbb{Q}$. In particular, we are interested
in the quantity $\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q})$ for fixed $E$
and varying $K$. When $\mathrm{rk}(E/\mathbb{Q}) \leq 1$, with $E$ subject to
some other conditions, we prove there are infinitely many $S_4$ quartic
extensions $K/\mathbb{Q}$ over which $E$ does not gain rank, i.e. such that
$\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q}) = 0$. 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 $\mathbb{Q}$.
