I think this is Jakke by Hannah Davenport. It was originally released in 2016, then rereleased in 2020 so the rerelease fits with the time you remember reading it.
The Tureis attacked Earth, killing most people in their first attacks. Sydney found a way to survive, carefully avoiding anything alien. When she found an injured Azziarin lying in a dark alley, she went against her every instinct and rescued him. After nursing him back to health, the Azziarin’s stormed her hideout, injuring her in the process.
Unable to let the injured female remain lying on the ground, Takkeo ordered that she be brought back to their ship. She had helped him and he couldn’t leave her injured. How was he supposed to know that his older brother and Commander of the ship would experience the mating fever with her presence?
Jakke had three consorts back home. Cassia had always thought that she would be his mate. He tried to avoid the earth female that brought forth his mating fever, but he ended up mated to her anyways. Now he had to explain things to her, his consorts, and his father, King Mallik. Could they all live together in unison? He sure hoped so.
The bad and good aliens are described as the gray aliens and the white haired aliens respectively.
Sydney finds the "good alien" boy, Takkeo, injured and rescues him:
She’d only made it around the corner and halfway down the alley when she stopped short—there was a person sprawled out like a dead body in the alley. She teetered. The last thing she wanted was to get involved, but how in good conscience could she walk away? What if he was alive and died because Sydney had been too scared to help?
Crap!
To make matters even more difficult, she knew he wasn’t Human. The white hair and dark skin marked him as possibly one of the good guys (such was yet to be determined). The only thing she knew about the aliens for certain was that the gray, pasty-skinned guys with red hair were definitely bad. Sydney would never help the gray guys, period.
Takkeo is rescued by his brother's officer Sagge. Sydney is injured in the attack and Takkeo insists she is rescued as well in return for helping him:
“Takkeo, are you okay?” Sagge asked as he knelt in front of the couch.
“Yes. Make sure the female is okay, and don’t let anyone hurt her!” Takkeo ordered.
Sagge looked over his shoulder at the female lying in a heap on the floor.
“No one touch the Human,” he ordered, although she was of no concern to them. “And notify the Commander that we have Takkeo.”
“She comes with us,” Takkeo said firmly.
Takkeo's elder brother Jakke then experiences mating fever as soon as he sees Sydney:
Rykker wasn’t making him feel any better. This was an impossible situation. The Human female set his body on fire in the most primal way. Her exotic looks combined with that petite frame forced protective instincts to kick in. And he had only been in her presence a handful of minutes…what would happen over the next three weeks?
And the rest I'll leave to your imagination.
