${\text{S}}_{\text{N}}1$ reaction occurs through the formation of carbocation as intermediate. Hence, rearrangement occurs in ${\text{S}}_{\text{N}}1$ reaction and not in ${\text{S}}_{\text{N}}2$ reaction. ${\text{S}}_{\text{N}}1$ reaction is favoured in presence of a polar protic solvent and a weak nucleophite. These reactions can be catalysed by the some Lewis is acid. A strong nucleophile is an aprotic solvent increases the rate of ${\text{S}}_{\text{N}}2$ reaction. Hence, option $\left(\text{B}\right)$ in incorrect.