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The first aim of this note is to fill a gap in the literature by provingthat, given a global field $K$ and a finite set $\mathcal{S}$ of primes of $K$,every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/K)$ over $K$with nilpotent kernel has a solution ${\rm{Gal}}(F/K) \rightarrow G$ such thatall primes in $\mathcal{S}$ are totally split in $F/L$. We then apply this toinverse Galois theory over division rings. Firstly, given a number field $K$ oflevel at least $4$, we show that every finite solvable group occurs as a Galoisgroup over the division ring $H_K$ of quaternions with coefficients in $K$.Secondly, given a finite split embedding problem with nilpotent kernel over afinite field $K$, we fully describe for which automorphisms $\sigma$ of $K$ theembedding problem acquires a solution over the skew field of fractions $K(T,\sigma)$ of the twisted polynomial ring $K[T, \sigma]$.

A note on finite embedding problems with nilpotent kernel