The unstable Adams spectral sequence for two-stage towers

Let KP denote a generalized mod 2 Eilenberg–MacLane space and let Y be the fiber of a map X→KP to which the Massey–Peterson theorem applies. We study the relationship of the mod 2 unstable Adams spectral sequence (UASS) for X and for Y. Given conditions on X, we split the E2-term for Y, and we use a primary level calculation to compute d2 for Y up to an error term. If the UASS for X collapses at E2 (for example, if X is an Eilenberg–MacLane space), the UASS for Y collapses at E3, and we have the entire UASS for Y. We also give examples and address a conjecture of Bousfield on the UASS for the Lie group SO.