Below is the mechanism from this link of the Beckmann Rearrangement.
In the alkyl migration step, the alkyl group migrates simultaneously as the $\ce{H2O}$ leaving group is expelled. This seems as though it is an $\mathrm{S_N2}$ reaction at the $\mathrm{sp^2}$-hybridized nitrogen atom, but I thought that it was geometrically impossible to have an $\mathrm{S_N2}$ reaction at an $\mathrm{sp^2}$ center, per this link.
What am I missing in my reasoning?


\ce{…}expression in formatting other than chemical equations and formulas. – Mathew Mahindaratne May 28 '20 at 18:33