No endorsement is needed. The AGI and IGI written tests can be done without anyone's endorsement.
- You also don't need an endorsement to take the CFI, CFII, or FOI written tests, btw.
- Your reference to 61.193 has nothing to do with this. For one, it's 61.193, not 61.139 (but your link itself is correct), and second, these are not endorsements for written tests. In 61.193 it says "to train and issue endorsements that are required for: ..." and that does not include any endorsements for the AGI, IGI, FIA, FII, and FOI written tests. Such endorsements, as they pertain to a ground instructor, would, for example, be a 61.217(d) endorsement (not written test).
- "... flight instructor privileges include providing an endorsement to take the test." No, they do not. They include providing an endorsement for a ground instructor certificate/licence (not written test), and endorsements for written tests (in general -- not for the ground instructor).
When an endorsement for the written test is needed, then that prerequisites say that explicitly, such as for the PPL, IR, CPL, ATPL, etc., such as in 61.65.(a)(4), for example.