It depends. If you generate rental income from that summer home, it is taxed to Georgia. Likewise, if you work in Georgia (whether or not you are staying at the summer home), the income is taxed in Georgia as a nonresident. Otherwise, your income is not taxable to Georgia because that summer home is not your domicile, or main home. Likewise, you are not a part-year resident for the same reason: your summer home is not your domicile.
However, you may still claim the mortgage interest and the real estate taxes you pay on this home as itemized deductions (unless you rent it also, in which case a percentage gets deducted for the rental use and the rest included with itemized for the personal use). Unless you were in Georgia for over half the year (summer home suggests that this is not the case), you claim no residency in Georgia.
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