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The theory of bisets has been very useful in progress towards settling thelongstanding question of determining units for the Burnside ring. In 2006 Boucused bisets to settle the question for $p$-groups. In this paper, we provide astandard basis for the unit group of the Burnside ring for groups that containa abelian subgroups of index two. We then extend this result to groups $G$,where $G$ has a normal subgroup, $N$, of odd index, such that $N$ contains anabelian subgroups of index $2$. Next, we study the structure of the unit groupof the Burnside ring as a biset functor, $B^\times$ on this class of groups anddetermine its lattice of subfunctors. We then use this to determine thecomposition factors of $B^\times$ over this class of groups. Additionally, wegive a sufficient condition for when the functor $B^\times$, defined on a classof groups closed under subquotients, has uncountably many subfunctors.

On the unit group of the Burnside ring for some solvable groups as a biset functor