Remove fancyhdr footer when float page too large

I'm working on a book with fancyhdr with some overfull float pages, and want to remove the footer on only those pages, but keep the header.

Can I either automatically remove footer if the caption writes over it or include something in the caption which does that?

\documentclass[a4paper,12pt]{book}
\usepackage{fancyhdr}

\pagestyle{fancyplain}
\fancyhf{}
\cfoot{\thepage}

\begin{document}
\mainmatter

\chapter{Introduction}
\section{First}
text

\begin{figure}[p]
\rule{1cm}{1cm}
\caption{small figure page, print page number}
\end{figure}

\begin{figure}[p]
\rule{10cm}{20.8cm}
\caption{large figure page, don't print page number}
\end{figure}

\end{document}


PS: Can I deactivate the "float too large" warning if it fits on the page once the footer is removed?

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I can't offer a solution for fancyhdr, but the recently released titleps package features (among other things) the commands \nextfloathead and \nextfloatfoot which force a header/footer for the float immediately following these commands.

Note that titleps uses a pagestyle definition syntax rather different from those of fancyhdr and the scrpage2 package -- see section 2 of the titleps manual for details, particularly about the \sethead and \setfoot commands.

\documentclass[a4paper,12pt]{book}

\usepackage[psfloats]{titleps}

{}{}{\textsl{\thesection~\sectiontitle}}%
\setfoot{}{\thepage}{}%
}
\renewpagestyle{plain}{%
\setfoot{}{\thepage}{}%
}

\begin{document}

\chapter{Introduction}

\section{First}

text

\begin{figure}[p]
\rule{1cm}{1cm}
\caption{small figure page, print page number}
\end{figure}

\nextfloatfoot{}{}{}{}

\begin{figure}[p]
\rule{10cm}{20.8cm}
\caption{large figure page, don't print page number}
\end{figure}

\end{document}

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