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In this paper we establish a formula for computing $d_2(sq^i(x))$ where $x$is a permanent cycle in the $C_2$-equivariant Adams spectral sequence or themotivic Adams spectral sequence over $Spec(\mathbb{R})$.  This requires establishing that the Adams towers have an$H_{\infty}$-structure as well as determining the attaching maps for$C_2$-equivariant projective spaces.  The attaching maps of $C_2$-equivariant projective spaces can then be used todetermine the coefficients of differentials in both the equivariant and motiviccase.  At the end some sample computations are given.

Squaring operations in the $RO(C_2)$-graded and real motivic Adams spectral sequences