One way:

/A/g;
s/\G(.*?)A/$1a/g;
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Another way:

s{(?<=A)(.*)}{my $rest=$1; $rest=~tr/A/a/; $rest}e;
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Without regex:

substr($_, index($_, 'A')+1) =~ tr/A/a/;
[download]

And, combining techniques:

/A/g;
substr($_, ++pos) =~ tr/A/a/;
[download]