We introduce the Invariant Ideal Axiom (IIA) , sketch a proof of its consistency and exhibit some of its consequences, the main one being:

Main Theorem: (IIA) Every countable sequential topological group is either metrizable or $k_\omega$.

Recall that a topological space $X$ is $k_\omega$ if there is a countable family $\{K_n: n\in\omega\}$ of compact subsets of $X$ such that a set $F\subseteq X$ is closed if and only if $F\cap K_n$ is closed for every $n\in\omega$. In particular, assuming IIA every countable sequential group topology is analytic (in fact, $F_{\sigma\delta}$).