This paper introduces the concept of super pseudo-amenable Banach algebras, which extends the established notion of pseudo-amenability. The study focuses on the super pseudo-amenability of Banach algebras associated with locally compact groups, particularly group algebras. Specifically, it is demonstrated that L 1 ( G ) is super pseudo-amenable if and only if G is an amenable [ S I N ] -group. Additionally, the super pseudo-amenability of certain inverse semigroup algebras is characterized. As an application, it is shown that for the Brandt semigroup S = M 0 ( G , I ) over a non-empty set I , the algebra ℓ 1 ( S ) is super pseudo-amenable if and only if G is amenable and I is finite.
