This is more of an extended comment than an answer. I will determine all the abelian groups failing to act faithfully on at most $n/3$ points.

Suppose $G$ is abelian, say $G = C_{q_1} \times C_{q_2} \times \cdots \times C_{q_k}$, where $q_1, q_2, \dots$ are (not necessarily distinct) prime powers. Then the minimal faithful permutation action of $G$ has $q_1 + q_2 + \cdots + q_k$ points. For a proof see https://mathoverflow.net/a/409831/20598.

Hence $G$ does not act faithully on $|G|/3$ points iff $$q_1 + \cdots + q_k > q_1 \cdots q_k / 3.$$ It is maybe easier to think of this as $$ \sum_{i=1}^k q_i / (q_1 \cdots q_k) > 1/3 . $$ Each of the terms here is at most $1/2^{k-1}$, so $k / 2^{k-1} > 1/3$, so $k \leq 4$. By further analyzing $k=1,2,3,4$ we find all the solutions (where $q_1 \leq \cdots \leq q_k$): $$ (q_1),\\ (2, q_2), (3, q_2),\\ (4, 4), (4, 5), (4, 7), (4, 8), (4, 9), (4, 11), (5, 5), (5, 7),\\ (2, 2, q_3) \qquad (2 \leq q_3 \leq 11),\\ (2, 3, 3), (2, 3, 4), (2, 3, 5), (2, 2, 2, 2), (2, 2, 2, 3). $$

This is more of an extended comment than an answer. I will determine all the abelian groups failing to act faithfully on at most $n/3$ points.

Suppose $G$ is abelian, say $G = C_{q_1} \times C_{q_2} \times \cdots \times C_{q_k}$, where $q_1, q_2, \dots$ are (not necessarily distinct) prime powers. Then the minimal faithful permutation action of $G$ has $q_1 + q_2 + \cdots + q_k$ points. For a proof see https://mathoverflow.net/a/409831/20598.

Hence $G$ does not act faithully on $|G|/3$ points iff $$q_1 + \cdots + q_k > q_1 \cdots q_k / 3.$$ It is maybe easier to think of this as $$ \sum_{i=1}^k q_i / (q_1 \cdots q_k) > 1/3 . $$ Each of the terms here is at most $1/2^{k-1}$, so $k / 2^{k-1} > 1/3$, so $k \leq 4$. By further analyzing $k=1,2,3,4$ we find all the solutions (where $q_1 \leq \cdots \leq q_k$): $$ (q_1),\\ (2, q_2), (3, q_2),\\ (4, 4), (4, 5), (4, 7), (4, 8), (4, 9), (4, 11), (5, 5), (5, 7),\\ (2, 2, q_3) \qquad (2 \leq q_3 \leq 11),\\ (2, 3, 3), (2, 3, 4), (2, 2, 2, 2), (2, 2, 2, 3). $$