The set of translation planes with spreads in P G(3, q) admitting at least three homology groups with distinct axes of order q + 1 is completely determined. Apart from the Desarguesian and Hall planes of order q 2 , the only possible plane is the Heimbeck plane of order 7 2 admitting several quaternion homology groups of order 8. A classification is also given of all translation planes with spreads in P G(3, q) that admit at least three distinct homology groups of order q + 1. Recent results

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... necting translation planes with spreads in P G(3, q) admitting cyclic affine homology groups of order q + 1 with conical flocks spreads provide the background for applications showing how the associated collineation groups are interrelated.