In 1984, Andrews introduced the family of partition functions \(cφ_k(n)\), which counts the number of generalized Frobenius partitions of \(n\) with \(k\) colors. In previous work, we proved a conjecture on congruences for \(cφ_6(n)\) modulo powers of 3. In this paper, we consider the \((6,0)\)-colored Frobenius partition functions \(cψ_{6,0}(n)\). We establish a connection between the generating functions of \(cψ_{6,3}(n)\) and \(cψ_{6,0}(n)\) via an Atkin-Lehner involution, and prove congruences modulo powers of 3 for \(cψ_{6,0}(n)\).