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We show that the N=2 superextended 1D quantum Dirac delta potential problemis characterized by the hidden nonlinear $su(2|2)$ superunitary symmetry. Theunexpected feature of this simple supersymmetric system is that it admits threedifferent $\mathbb Z_2$-gradings, which produce a separation of 16 integrals ofmotion into three different sets of 8 bosonic and 8 fermionic operators. Thesethree different graded sets of integrals generate two different nonlinear,deformed forms of $su(2|2)$, in which the Hamiltonian plays a role of amultiplicative central charge. On the ground state, the nonlinear superalgebrais reduced to the two distinct 2D Euclidean analogs of a superextendedPoincar\'e algebra used earlier in the literature for investigation ofspontaneous supersymmetry breaking. We indicate that the observed exoticsupersymmetric structure with three different $\mathbb Z_2$-gradings can beuseful for the search of hidden symmetries in some other quantum systems, inparticular, related to the Lam\'e equation.

Hidden nonlinear su(2|2) superunitary symmetry of N=2 superextended 1D Dirac delta potential problem