In most cases, extensions between Serre weights for unramified GL_3(O_L) modulo the center coincide with the corresponding GL_3(F_q) extensions.
[Her09] Florian Herzig
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On the $K$-extensions between Serre weights for unramified $\mathrm{GL}_3$
In most cases, extensions between Serre weights for unramified GL_3(O_L) modulo the center coincide with the corresponding GL_3(F_q) extensions.