Let

$M/N$ be a Frobenius extension of

$k$-algebras with Frobenius homomorphism

$E$ and dual bases

$\left\{{x}_{i}\right\}$,

$\left\{{y}_{i}\right\}$. Let

$U={C}_{M}\left(N\right)$. The extension is called symmetric if

$E$ commutes with every

$u\in U$, and Markov if the extension is strongly separable (i.e.

$E\left(1\right)=1$ and

${\sum}_{i}{x}_{i}{y}_{i}={\lambda}^{-1}1$) and there is a (Markov) trace

$T:N\to k$ such that

$T\left(1\right)={1}_{k}$ and

${T}_{0}=T\circ E:M\to k$ is a trace. The basic construction theorem says that if

$N\subseteq M$ is a symmetric Markov extension and

${M}_{1}=M{\otimes}_{N}M=\text{End}\left({M}_{N}\right)$ then

${M}_{1}/M$ is a symmetric Markov extension; the Frobenius endomorphism

${E}_{M}$ and the dual bases are described, and the Markov trace is

${T}_{0}$. If in addition

$U$ is Kanzaki separable,

${T}_{0}{|}_{U}$ is non-degenerate and

${\sum}_{i}{x}_{i}{y}_{i}={\sum}_{i}{y}_{i}{x}_{i}$ then

$V={C}_{{M}_{1}}\left(M\right)$ is Kansaki separable and the restriction of

${T}_{1}={T}_{0}\circ {E}_{M}$ to

$V$ is non-degenerate. The construction can be iterated to obtain the Jones tower

$N\subseteq M\subseteq {M}_{1}\subseteq {M}_{2}$. Let

$A={C}_{{M}_{1}}\left(N\right)$ and

$B={C}_{{M}_{2}}\left(M\right)$. Assuming the existence of dual bases for

${E}_{M}$ resp.

${E}_{{M}_{1}}$ in

$A$ resp.

$B$ (depth 2 condition) the authors prove some properties of algebra extensions involving

$A$,

$B$,

$U$ and

$V$. Examples of extensions with depth 2 are provided in the final appendix. Then semisimple weak Hopf algebra structures are defined in

$A$ and

$B$, and also

$A$ resp.

$B$-module algebra structures on

$M$ resp.

${M}_{1}$. Two isomorphisms

${M}_{1}\simeq M\#A$ and

${M}_{2}\simeq {M}_{1}\#B$ are also provided. Finally, in the absence of a trace, the basic construction is iterated to the right, extending a result of

*M. Pimsner* and

*S. Popa* [Trans. Am. Math. Soc. 310, No. 1, 127-133 (1988;

Zbl 0706.46047)].