Finite quasiprimitive permutation groups of twisted wreath type are the finite permutation groups with a unique minimal normal subgroup which is non-abelian, non-simple and acts regularly. If T is a non-abelian simple group and P is a group that conveys transitive action on the set \(\textbf{k}=\{1,2,\ldots ,k\}\) with \(k\geqslant 2\) , then every permutation group in this classification can be considered permutation isomorphic to \(G=T^k{:}P\) , a twisted wreath product acting on its base group \(\Omega =T^k\) . We prove that if \(T\cong \textrm{A}_n,P\cong M^l{:}N\leqslant \textrm{S}_k\) with \(M=\textrm{A}_s\) or classical group with dimensions less than or equal to \(n-2\) , \(n\leqslant \{8,s,\ell \}\) , then the base size of G is 2. Additionally, we demonstrate three possible values of the base size when P is semiprimitive on \(\textbf{k}\) and G is quasiprimitive on \(\Omega \) .
