It's a 3 month internship at this point so I'm really not certain as to how to go about this the way you've laid it out here.
That's the thing, there's no particular one way to go about it. If you're employed the rest of the year, you can probably just bump up your employer withholdings during that time and not worry about the self-employed months until you file your taxes next year. If you don't know what the rest of the year holds, then you'll want to pay as you go to avoid penalties.
If you're not sure, I'd recommend saving about 1/4 of your salary to cover tax payments and filling out the estimated tax form for whichever quarters apply. Please note that that's a ROUGH estimate and I'm not an accountant.
Congrats on the internship BTW, I forgot to say that before.
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