The Halacha prohibits instructing a non-Jew on Shabbat to perform a Melacha forbidden from the Torah, even for the purpose of a Misva. Therefore, one may not tell a non-Jew to turn on the lights to learn Torah on Shabbat. However, the exception to this rule is telling the non-Jew during Ben HaShmashot (twilight-defined as the 13.5 minutes after sunset).
For example, if the Shabbat hot plate was not plugged in, a non-Jew may be utilized during that period of time, even though the woman has already lit candles and accepted Shabbat. During Ben HaShmashot, any Melacha, even If prohibited by Torah law, may be performed by the non-Jew for the sake of a Misva.
This principle applies to the twilight at the end of Shabbat, as well. For example, if sunset is at 6 PM and Shabbat concludes 42 minutes later, at 6:42 PM, and the lights went out in the synagogue during that period, a non-Jew may be called to turn them on, since it is for the sake of a Misva.
SUMMARY
During Ben HaShmashot of Ereb Shabbat or Mosa'eh Shabbat, any Melacha, even If prohibited by Torah law, may be performed by a non-Jew for the sake of a Misva.
