On the surjectivity of the power maps of a class of solvable groups

Let $G$ be a group containing a nilpotent normal subgroup $N$ with central series $\{N_j\}$, such that each $N_j/N_{j+1}$ is a $\mathbb{F}$-vector space over a field $\mathbb{F}$ and the action of $G$ on $N_j/N_{j+1}$ induced by the conjugation action is $\mathbb{F}$-linear. For $k\in \mathbb N$ we describe a necessary and sufficient condition for all elements from any coset $xN$, $x\in G$, to admit $k$-th roots in $G$, in terms of the action of $x$ on the quotients $N_j/N_{j+1}.$ This yields in particular a condition for surjectivity of the power maps, generalising various results known in special cases. For $\mathbb{F}$-algebraic groups we also characterise the property in terms of centralizers of elements. For a class of Lie groups, it is shown that surjectivity of the $k$-th power map, $k\in \mathbb N$, implies the same for the restriction of the map to the solvable radical of the group. The results are applied in particular to the study of exponentiality of Lie groups.