Classical linear ciphers, such as the Hill cipher, operate on fixed, finite-dimensional modules and are therefore vulnerable to straightforward known-plaintext attacks that recover the key as a fully determined linear operator. We propose a symmetric-key cryptosystem whose linear action takes place instead in the Burnside ring $A(G)$ of a compact Lie group $G$, with emphasis on the case $G=O(2)$. The secret key consists of (i) a compact Lie group $G$; (ii) a secret total ordering of the subgroup orbit-basis of $A(G)$; and (iii) a finite set $S$ of indices of irreducible $G$-representations, whose associated basic degrees define an involutory multiplier $k\in A(G)$. Messages of arbitrary finite length are encoded as finitely supported elements of $A(G)$ and encrypted via the Burnside product with $k$. For $G=O(2)$ we prove that encryption preserves plaintext support among the generators $\{(D_1),\dots,(D_L),(SO(2)),(O(2))\}$, avoiding ciphertext expansion and security leakage. We then analyze security in passive models, showing that any finite set of observations constrains the action only on a finite-rank submodule $W_L\subset A(O(2))$, and we show information-theoretic non-identifiability of the key from such data. Finally, we prove the scheme is \emph{not} IND-CPA secure, by presenting a one-query chosen-plaintext distinguisher based on dihedral probes.