On certain near-domains and sharply $2$-transitive groups

We find sufficient conditions under which a near-domain is a near-field and a $2$-transitive group has a normal regular abelian subgroup. If a sharply $2$-transitive group $T$ ($\mathrm{Char}\,T\ne2$) contains a Frobenius group with involution such that its complement contains a subgroup of order $>2$ that is normal in the stabilizer of a point, then $T$ has a regular abelian normal subgroup (Theorem 1). If, in a near-domain of odd characteristic, there is a near-field containing a multiplicative subgroup of order $>2$ that is normal in a multiplicative group of the near-domain, then the near-domain is a near-field (Theorem 2). This result also holds in the case when the local nilpotent radical of the stabilizer of a point contains a $2$-subgroup of order $\geq16$ and the characteristic is congruent to 1 modulo 16 (Theorem 3).