  We introduce a Floer homotopy version of the contact invariant introduced by
Kronheimer-Mrowka-Ozv\'ath-Szab\'o. Moreover, we prove a gluing formula
relating our invariant with the first author's Bauer-Furuta type invariant,
which refines Kronheimer-Mrowka's invariant for 4-manifolds with contact
boundary. As an application, we give a constraint for a certain class of
symplectic fillings using equivariant KO-cohomology.
