The Stefan-Boltzmann law for net power radiated pertains to the object. That is, we're simply asking, how much radiation leaves this object (this depends on the object's emissivity), and how much radiation is absorbed by this object (this depends on the objects absorptivity). The emissivity and absorptivity in the equation you present thus pertain to the object, not the environment. That equation makes some assumptions. I couldn't find a good explanation for why the coefficients are what they are in the net power formula you posted, so I thought I'd take a step back and derive it.
The power emitted per unit area from the surroundings is
$$P_s=\epsilon_s \sigma T_s^4$$
The object will absorb a fraction of that based on its area and absorptivity:
$$P_a=\alpha \epsilon_s \sigma T_s^4$$
The object will emit:
$$P_e=\epsilon \sigma T^4$$
The net power delivered to the object is
$$P_{net} = P_a - P_e = \epsilon\sigma T^4 - \alpha \epsilon_s \sigma T_s^4$$
If the absorptivity and emissivity are equal, and $\epsilon_s = 1$ (blackbody), we get:
$$P_{net} = P_a - P_e = \epsilon \sigma (T^4-T_s^4)$$
So you'd have to assume that the surroundings perfectly emitting, and that the absorptivity and emissivity are equal. The latter is true under thermodynamic equilibrium or local thermodynamic equilibrium. See the Wikipedia page for Planck's law and in particular the section on Kirchhoff's Law.
UPDATED:
I now think my previous answer was wrong, because the set up would be equivalent to the following question: Is a black body sphere inside a black body shell hotter than the shell?
Just change the question to add a carefully crafted lens that focuses all the radiation into the sphere (you could make the shell as large as you want), which of course is impossible to make or it would violate the second law.
Best Answer
Not all the radiation from the outer shell reaches the inner shell. When you take into account the intensity distribution of radiation from the outer shell (Lambertian distribution, i.e. $\propto\cos\theta$) you will see that the amount of radiation for the inner to the outer shell is the same as in the other direction.
No violation of the second law.