  By considering a particular type of invariant Seifert surfaces we define a
homomorphism $\Phi$ from the equivariant concordance group of directed strongly
invertible knots $\widetilde{\mathcal{C}}$ to a new equivariant algebraic
concordance group $\widetilde{\mathcal{G}}^{\mathbb{Z}}$. We prove that $\Phi$
lifts both Miller and Powell's equivariant algebraic concordance homomorphism
and Alfieri and Boyle's equivariant signature. Moreover, we provide a partial
result on the isomorphism type of $\widetilde{\mathcal{G}}^{\mathbb{Z}}$ and we
obtain a new obstruction to equivariant sliceness, which can be viewed as an
equivariant Fox-Milnor condition. We define new equivariant signatures and
using these we obtain novel lower bounds on the equivariant slice genus.
Finally, we show that $\Phi$ is invariant under topological equivariant
concordance and that can obstruct equivariant sliceness for knots with
Alexander polynomial one.
