Do External Auditors Should rely on work of internal auditors?

To some extent, Should be..But external auditor should apply own auditing procedures to reach at reliable level. Substantive Procedures are used to perform an external audit.

Yes , they do rely on internal auditor.   

Depends. External auditors should see to it that the internal auditor is exercising independence in doing his job. External auditors should be skeptic  to validate their works internal auditor.

Much of the work performed by a company’s internal audit function can overlap with the work conducted by the external auditor, specifically in areas dealing with the assessment of control processes. It is likely that in carrying out detailed work evaluating and reviewing the company’s internal control framework internal audit perform procedures on financial controls relevant to the external audit. As such, the external auditor, rather than duplicating these procedures, may be able to place reliance on the work carried out by the internal auditor. The International Standards On Auditing (ISAs) now highlight three ways in which internal auditors may be utilized by the external auditor in the audit of financial statements:

For sure. the reson is the IA is reported to Audit committee and he has no insterest to not doing his job perfectly But external auditor should apply own auditing procedures to reach at reliable level

External auditors normally rely the internal audit reports for having an idea on the extent of control the mgt had over the laid down procedures of accounting principles.At the same time, they will go thr' the accounts or will keep a tab on the standards to be followed while finalizing the accounts.Since they are normally appointed externally,are free to filter inappropriate entries and give a true picture of the state of affairs.

Well it is the judgment skills of external auditors. If they think that internal auditors are working fine then they can rely on them to a certain extent. However they cannot write in the audit report that they have relied on internal auditors work.