Is the article "a" justified in "they are wearing a cape"

You can't cast aspersions on someone just because they are wearing a cape.

Shouldn't it be capes instead of a cape? "a cape" means one cape, "they are" implies it is a group of people, a group of people can't wear one cape.

The only way I see it grammatically correct is when they stands for "he or she" to avoid sexism. Like in the sentence I saw in one video game: "The player doofos killed you with their AK47."

Answer

You actually have the answer to your question, within your question.

Let me split the sentence : You can't cast aspersions/ on someone/ just because /they are/ wearing a cape.

Note: I have not split this sentence according to any grammatical convention, I just split it up for the purpose of the discussion.

• 
  

  In the sentence you are talking about "someone" and well "someone " is singular, so your sentence is talking about a general, unknown him/her.

  
  


• 
  

  Since we are talking about a singular him/her, the usage "a cape" is correct.

  
  


• 
  

  Now, how does "they are" which implies a group of people fit into this sentence? Well let me introduce you to 'Singular they '. It is somewhat similar to the usage of 'you'
  
  eg: you are strong - Singular reference
  you guys stink - referring to a group of people but it is the same 'you'

  
  


• Similarly 'they' can be used in singular and plural forms, and in your question, they is singular. Talking of singular they, I remember an example,
  when we were kids, we used to make fun of our teacher behind their back and giggle. Every time this happens, the teacher would shout " If anybody thinks they are so smart, you teach the class I will listen!" (huh, fun times).


• Well in the above sentence 'they' is used in singular context again.

If you want to become a "Singular They Expert" you gotta check this out.

There is another more complicated concept to this "Singular They" called as epicene but explore at your own risk. I dont want do discuss further about epicene as I am not very knowledgeable with the grammatical nuances of an epicene.

Edit: I agree with @sgryzko 's comment, maybe the example I originally gave needs a little modification. The example suggested by @mplungjan is definitely an improvement - "When we were kids, we used to make fun of each and every teacher behind their back and giggle".