Between the two options, the correct one is 'sitting'. 'Making' would be wrong.
In fact, the more usual wording would be 'Anna is taking the First Certificate in June'. In the English-speaking world as a whole, it is far more common to say 'take a test'.
As Angela has pointed out, 'sit' sounds unnatural to the American ear. American English sometimes uses 'write', but that sounds [very very very :) ] wrong to GBE speakers, so it would be best for you to use the universally acceptable 'take'.
In Britain, 'sit an exam' is used, but it is quite formal and a little old-fashioned. An official communication from an exam board might use 'sit', but the students themselves would be more likely to say 'take'.
On a different point...may I correct your question? I knew before I opened your page or looked at your profile that you were a Russian speaker. How? Because of your incorrect use of the word 'variant'. This mistake is so common among Russian speakers that I am sure that it is being taught. Variants are equally acceptable versions of the same basic thing - so you can't use it to refer to two different words, one of which is wrong. The word you need here is 'options', not variants.
I hope that helps.