"Each word has W as the second letter" is a lame connection. I figured this category to be a word problem, and I guessed "each contains another word" (twee[t], sweat[er], [a]ward, own[er]) -- which fits and was even accepted merely because I'd used the word "word" in my answer. Even with that, I felt surely I was grasping at straws. But a certain letter recurs in a certain position? That really is grasping at straws for puzzle setting!