Characterization of saturated sets containing no even numbers

Let $G$ be the abelian group under discussion, and call a subset saturated if it is a maximal avoidable set. The sets described above are those obtained as solution sets of

$$\sum_{i=1}^n a_i x_i\equiv 1\pmod{2},$$

where each $a_i$ is $0$ or $1$, not all $0$, in the decomposition of $G$ given in the preceding context. Characterization conjecture. The sets described above are the only saturated sets containing no even numbers. This would classify all saturated sets avoiding even elements; the source does not provide evidence resolving the assertion.