Abelian extensions of topological vector groups

The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group $\text{G}\text{F}$ are topological vector groups is investigated. It is shown that, if F is a real Fréchet group and $\text{G}\text{F}$ a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if $\text{G}\text{F}$ is finite dimensional or if F is one dimensional and $\text{G}\text{F}$ a separable Hilbert group.