Each topological group embeds into a duoseparable topological group

A topological group $X$ is called $duoseparable$ if there exists a countable set $Ssubseteq X$ such that $SUS=X$ for any neighborhood $Usubseteq X$ of the unit. We construct a functor $F$ assigning to each (abelian) topological group $X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an isomorphic copy of $X$. In fact, the functor $F$ is defined on the category of unital topologized magmas. Also we prove that each $sigma$-compact locally compact abelian topological group embeds into a duoseparable locally compact abelian-by-countable topological group.