  We establish a surgery exact triangle for involutive Heegaard Floer homology
by using a doubling model of the involution. We use this exact triangle to give
an involutive version of Ozsv\'ath-Szab\'o's mapping cone formula for knot
surgery. As an application, we use this surgery formula to give examples of
integer homology spheres that are not homology cobordant to any linear
combination of Seifert fibered spaces.
