Counterhate arguments can effectively fight and limit the spread of hate speech. However, they can also exacerbate the hate, as some people may respond with aggression if they feel threatened or targeted by the counterhate. In this paper, we investigate replies to counterhate arguments beyond whether the reply agrees or disagrees with the counterhate argument. We present a corpus with 2,621 replies to counterhate arguments countering hateful tweets, and annotate them with fine-grained characteristics. We show that (a) half of the replies (51%) to the counterhate arguments disagree with the argument, and (b) this kind of reply often supports the hateful tweet (40%). We also analyze the language of counterhate arguments that elicit certain types of replies. Experimental results show that it is feasible to anticipate the kind of replies a counterhate argument will elicit.