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We show that if we factor the long exact sequence in cohomology of a cofibersequence of spectra into short exact sequences, then the $d_2$-differential inthe Adams spectral sequence of any one term is related in a precise way toYoneda composition with the 2-extension given by the complementary terms in thelong exact sequence. We use this to give a complete analysis of the Adamsspectral sequence for the connective image-of-$J$ spectrum, finishing acalculation that was begun by D. Davis in 1975.

The Adams spectral sequence for the image-of-$J$ spectrum