Subgroup of Index 3 does not necessarily contain all Cubes of Group Elements

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$ whose index is $3$.

Then it is not necessarily the case that:
 * $\forall x \in G: x^3 \in H$

Proof
Proof by Counterexample:

Consider $S_3$, the symmetric group on $3$ letters.

From Subgroups of Symmetric Group on 3 Letters, the subgroups of $S_3$ are:

subsets of $S_3$ which form subgroups of $S_3$ are:

One such subgroup of $G$ whose index is $3$ is $\set {e, \tuple {12} }$

But $\set {e, \tuple {12} }$ does not contain $\tuple {123}$ or $\tuple {132}$, both of which are of order $3$.