On equivalence relations second order definable over H(κ)

Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 second order definable over H(κ) if there exists a second order sentence φ and a parameter P ⊆ H(κ) such that functions f and g from κ into 2 are equivalent iff the structure 〈H(κ),∈, P, f, g〉 satisfies φ. The possible numbers of equivalence classes of second order definable equivalence relations contains all the nonzero cardinals at most κ. Additionally, the possibilities are closed under unions and products of at most κ cardinals. We prove that these are the only restrictions: Assuming that GCH holds and λ is a cardinal with λ = λ, there exists a generic extension, where all the cardinals are preserved, there are no new subsets of cardinality < κ, 2 = λ, and for all cardinals μ, the number of equivalence classes of some second order definable equivalence relation on functions from κ into 2 is μ iff μ is in Ω, where Ω is any prearranged subset of λ such that 0 6∈ Ω, Ω contains all the nonzero cardinals ≤ κ, and Ω is closed under unions and products of at most κ cardinals. 1