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– Copy over the header.php from bp-default theme to your child theme.
– Open header.php and add class to the li’s of default BP components starting from line 47 on header.php

<li class="icon-home">
<li class="icon-activity">
<li class="icon-members">
<li class="icon-groups">
<li class="icon-forums">

You would have to ask plugin authors of “Links” and “Achievements” on how to generate addition class for styling hook.
– after you’re done, save and upload to your child theme folder in server
– in style.css add background image to li.icon-members, li.icon–groups, etc.