You should not talk about an extension, if you use the core newletter system.
HTML for list and reader module in FE gets rendered by mod_newsletter_list and mod_newsletter_reader.
Mail content gets rendered by mail_default.
Values of newsletters are stored in DB tl_newsletter. You can check the content there first.
HTML markup for the mail is optimized for as many mail clients as possible. So you can not compare mail markup with FE markup.
In templates you can use a snippet like this to generate debug output files of template variables which you then will find in system/logs/:
PHP Code:
<?php
/** DEBUG */ // remove last slash from "*/" after debugging
$myVar = $this->arrData; // or $this or another variable to debug
$myTime = '-'.microtime(true);
$logFile = basename(__FILE__).'-debug'.$myTime.'.log';
$debug = "\n\n".'file: '.__FILE__.' (line '.__LINE__.')'."\n\n";
$debug .= print_r($myVar, true);
log_message(str_replace(' ', ' ', $debug), $logFile);
/** */
?>
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