How to merge multiple .fasta flies retaining the file headers

Hi anyone has an idea about merging multiple .fasta flies retaining the file headers? I tried cat command but it will only keep the 1st header.
What I want to do:

FileA

>Header1
Sequence1
>Header2
Sequence2

FileB

>Header3
Sequence3

FileC

>Header4
  Sequence4
>Header5
  Sequence5

What I want:

New File

>Header1
  Sequence1
>Header2
  Sequence2
>Header3
  Sequence3
>Header4
  Sequence4
>Header5
  Sequence5

Thank you so much for your help!