= Solution
Take $N$ to be a closed compact <Riemannian manifold> and use the <positive Laplace-Beltrami operator>. Its <Riemannian heat kernel> $K_N(t,x,y)$ is the integral kernel of $e^{-t\Delta_N}$:
$$
(e^{-t\Delta_N}f)(x)=\int_NK_N(t,x,y)f(y)\,dV(y).
$$
It solves $(\partial_t+\Delta_x)K_N=0$ for $t>0$ and tends to the identity kernel as $t\downarrow0$. For an <orthonormal eigenbasis>, the <spectral expansion of the Riemannian heat kernel> is
$$
K_N(t,x,y)=\sum_j e^{-t\lambda_j}\phi_j(x)\overline{\phi_j(y)}.
$$
The bar is required for a complex basis. With boundary, a specified invariant <boundary condition> must also be included in this definition.
For the quotient formula use the covering-space interpretation: $U$ acts freely and properly discontinuously by <Riemannian isometries>. On compact $N$ such a discrete group is finite. Freeness alone, without this covering hypothesis, does not justify the image sum: an infinite dense <subgroup> of circle rotations, for example, acts freely but has no manifold quotient. Positive-dimensional group actions require a different quotient analysis.
For lifts $x,y$ of $\bar x,\bar y\in M$, the <heat kernel on a finite isometric quotient> is
$$
\boxed{K_M(t,\bar x,\bar y)=\sum_{u\in U}K_N(t,x,uy).}
$$
Invariance of $K_N$ under simultaneous <Riemannian isometries> and reindexing the sum show that this is independent of both lifts. A local isometry commutes with the Laplacian, so the sum satisfies the quotient <heat equation>. For its initial condition, integrate over a fundamental domain $F\subset N$ against a lifted function. The terms combine into the integral over all of $N$, whose initial limit is $f(x)$. Uniqueness of the heat evolution proves the formula. \b[There is no factor $1/|U|$ in this kernel formula.]
On the diagonal the sum is $U$-invariant. Its integral over $F$ is therefore $1/|U|$ times its integral over $N$, giving the <heat trace>
$$
\boxed{Z_M(t)=\operatorname{Tr}(e^{-t\Delta_M})
=\frac1{|U|}\sum_{u\in U}\int_NK_N(t,x,ux)\,dV(x).}
$$
Thus the normalization factor appears in the trace, not the kernel. Put $F_t(g)=\int_NK_N(t,x,gx)\,dV(x)$. For any finite isometry group $T$, changing variables $x=hy$ proves $F_t(hgh^{-1})=F_t(g)$: it is a <class function> on $T$.
For <Gassmann equivalent> <subgroups> $U_1,U_2\leq T$, their intersections with each <conjugacy class> have equal size, and their orders are equal. If both act freely, the <heat traces> of their quotients satisfy
$$
Z_{U_i\backslash N}(t)=\frac1{|U_i|}\sum_{\mathcal C}|U_i\cap\mathcal C|F_t(\mathcal C),
$$
so they agree for every $t>0$. Since $Z_M(t)=\sum_\lambda m(\lambda)e^{-t\lambda}$, equality determines the <spectrum> with multiplicities: take $t\to\infty$ to recover the smallest <eigenvalue> and its multiplicity, subtract that term, and repeat. This proves <Sunada theorem>. Equivalently, projection onto $U_i$-invariant functions averages the group action, and the quotient <eigenvalue> multiplicity is $|U_i|^{-1}\sum_{u\in U_i}\chi_\lambda(u)$, with $\chi_\lambda$ the <eigenspace> <character of a representation>. Almost conjugacy equalizes these averages.
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